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16568883-Canonical-Structures-in-Potential-Theory-SS-Vinogradov-P-D-Smith-ED-Vino-Grad-Ova

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Downloaded copy of a published monograph (Part I of a two-volume work on potential theory and scattering), not Phil's own writing. It covers Laplace's equation in curvilinear coordinates, dual and triple series and integral equations, and regularisation via Abel transforms. Applications include electrostatics of open spherical, spheroidal, toroidal and conical shells, slotted cylinders, and flat plates, with appendices on special functions and functional analysis.

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©2001 CRC Press LLC To our children ©2001 CRC Press LLC Contents 1Laplace’ sEquatio n 1.1.1Cartesiancoordinates 1.1.2Cylindricalpolarc oordinates 1.1.3Sphericalpolarcoordinate s 1.1.4Prolatespheroidalc oordinates 1.1.5Oblatespheroidalcoordinates1.1.6Ellipticcylinde rcoordinate s 1.1.7Toroidalcoordinate s1.1Laplace’ sequatio nincurvilinea rcoordinate s 1.2.1Cartesiancoordinates1.2.2Cylindricalpolarc oordinates 1.2.3Sphericalpolarcoordinate s 1.2.4Prolatespheroidalc oordinates 1.2.5Oblatespheroidalcoordinates1.2.6Ellipticcylinde rcoordinate s 1.2.7Toroidalcoordinate s1.2Solution sofLaplace’ sequation :separatio nofvariable s 1.3Formulatio nofpotentialtheor yforstructure swithedges 1.4.1Thedeﬁnitionmethod1.4.2Thesubstitutionmethod1.4.3Noble’ smultiplyingfactormeth od 1.4.4TheAbelintegraltransfor mmeth od1.4Dualequations :aclassiﬁcatio nofsolutio nmeth ods 1.5Abel’sintegra lequatio nandAbelintegra ltransform s 1.6Abel-typeintegra lreprese ntation sofhypergeometri cfunction s 1.7Dualequation sandsingle -ordouble-l ayersurfac epotentials 2SeriesandIntegralEquations 2.1Dualseriesequation sinvolvin gJacob ipolynomial s 2.2Dualseriesequation sinvolvin gtrigonometrica lfunction s 2.3Dualseriesequation sinvolvin gassociatedLegendr efunction s 2.4.1TypeAtripleseriesequation s 2.4.2TypeBtripl eserie sequation s 2.4 Symmetric triple series equations involving Jacobi polynomials ©200 1 CRC Press LLC 2.5Relationship sbetweenseriesandintegra lequation s 2.6Dualintegra lequation sinvolvin gBesse lfunction s 2.7Nonsymmetrica ltripleseriesequation s 2.8Couple dseriesequation s 2.9Aclassofintegro-serie sequation s 3Electrostati cPotentialTheor yforOpenSpherica lShell s 3.1Theopenconductin gspherica lshell 3.2.1Appr oximat eanalytica lformulaeforcapacitanc e3.2Asymmetrica lpairofopenspherica lcapsandthespherical barre l 3.3Anasymmetrica lpairofspherica lcapsandtheasymmetric barre l 3.4Themeth odofinversion 3.5Electrostati cﬁeldsinaspherica lelectroni clens 3.6Frozenmagneti cﬁeldsinsidesuperconductin gshells 3.7Screenin gnumberofsuperconductin gshells 4Electrostati cPotentialTheor yforOpenSpheroida lShell s 4.1Formulatio nofmixedboundar yvalueproblem sinspheroidal geometr y 4.2Theprolat espheroida lconducto rwithonehole 4.3Theprolat espheroida lconducto rwithalongitudina lslot 4.4Theprolat espheroida lconducto rwithtwocircula rholes 4.5Theoblat espheroida lconducto rwithalongitudina lslot 4.6Theoblat espheroida lconducto rwithtwocircula rholes 4.7.1Openspheroida lshells 4.7.2Spheroida lcondensor s4.7Capacitan ceofspheroida lconductor s 5Charge dToroida lShell s 5.1Formulatio nofmixedboundar yvalueproblem sintoroida lge- ometr y 5.2Theopencharge dtoroida lsegme nt 5.3Thetoroida lshellwithtwotrans versalslots 5.4Thetoroida lshellwithtwolongitudina lslots 5.5Capacitanc eoftoroida lconductor s 5.6.1Thetoroida lshellwithoneazimuthalcut5.6 Anopentoroidal shell with azimuthal cuts ©200 1 CRC Press LLC 5.6.2Thetoroidalshel lwithmultiplecuts 5.6.3Limitingcases 6PotentialTheoryforConicalStructure swit hEdges 6.1Non-coplana roppositel ycharge dinﬁnit estrips 6.2Electrostati cﬁeldsofacharge daxisymmetri cﬁniteopenconi- calconducto r 6.3Theslotte dhollowspindl e 6.4Aspherica lshellwithanazimuthalslot 7Two-dimensiona lPotentialTheor y 7.1Thecircula rarc 7.2Axiall yslotte dopencircula rcylinder s 7.3Electrostati cpotentialofsystem sofcharge dthinstrips 7.4Axially-slotte dellipti ccylinder s 7.5Slotte dcylinder sofarbitrar yproﬁl e 8Mor eComplicatedStructures 8.1Rigorou ssolutio nmeth odsforcharge dﬂatplate s 8.2.1Thespherically-cur vedellipticplat e8.2Thecharge dellipti cplate 8.3Polygona lplate s 8.4Theﬁnitestrip 8.5Couple dcharge dconductors :thespherica lcapandcircula rdisc ANotation BSpecialFunction s B.1TheGamm afunctio n B.2Hypergeometri cfunction s B.3.1TheassociatedLegendr epolynomials B.3.2TheLegendr epolynomialsB.3Orthogona lpolynomials :Jacob ipolynomials ,Legendr epoly- nomial s B.4.1Ordinar yLegendr efunction sB.4 Associated Legend refunctions ©200 1 CRC Press LLC B.4.2Conicalfunction s B.4.3Ass ociatedLegendr efunction sofintegerorde r B.5.1SphericalBesse lfunction s B.5.2Modiﬁe dBesselfunctionsB.5Besse lfunction s B.6Theincomplet escalarproduct CElementsofFunctionalAnalysis C.1Hilbertspace s C.2Operator s C.3TheFredhol malternati veandregularisatio n DTransformsandIntegrationofSeries D.1Fourie randHankeltransform s D.2Integratio nofseries Reference s ©200 1 CRC Press LLC Preface Potential theory has its roots in the physical sciences and continues to ﬁnd application in diverse areas including electrostatics and elasticity. From amathematical point of view, the study of Laplace’s equation has profoundlyinﬂuenced the theory of partial diﬀerential equations and the development of functional analysis. Together with the wave operator and the diﬀusion opera- tor, its study and application continue to dominate many areas of mathemat-ics, physics, and engineering. Scattering of electromagnetic or acoustic wavesis of widespread interest, because of the enormous number of technological ap-plications developed in the last century, from imaging to telecommunicationsand radio astronomy. The advent of powerful computing resources has facilitated numerical mod- elling and simulation of many concrete problems in potential theory and scat-tering. The many methods developed and reﬁned in the last three decadeshave had a signiﬁcant impact in providing numerical solutions and insight intothe important mechanisms in scattering and associated static problems. How- ever, the accuracy of present-day purely numerical methods can be diﬃcult to ascertain, particularly for objects of some complexity incorporating edges,re-entrant structures, and dielectrics. An example is the open metallic cavitywith a dielectric inclusion. The study of closed bodies with smooth surfaces israther more completely developed, from an analytical and numerical point ofview, and computational algorithms have attained a good degree of accuracy and generality. In contradistinction to highly developed analysis for closed bodies of simple geometric shape – which was the subject of Bowman, Senior,and Uslenghi’s classic text on scattering [6] – structures with edges, cavities,or inclusions have seemed, until now, intractable to analytical methods. Our motivation for this two-volume text on scattering and potential theory is to describe a class of analytic and semi-analytic techniques for accurately de- termining the diﬀraction from structures comprising edges and other complex cavity features. These techniques rely heavily on the solution of associatedpotential problems for these structures developed in Part I. These techniques are applied to various classes of canonical scatterers, of particular relevance to edge-cavity structures. There are several reasons forfocusing on such canonical objects. The exact solution to a potential theory proble mordiﬀractio nproble misinterestin ginitsownright.AsBowmanet al. [6] state, most of our understanding of how scattering takes place is ob- tained by detailed examination of such representative scatterers. Their studyprovides an exact quantiﬁcation of the eﬀects of edges, cavities, and inclusions. ©200 1 CRC Press LLC This is invaluable for assessing the relative importance of these eﬀects in other, more general structures. Sometimes the solution developed in the text is inthe form of a linear system of equations for which the solution accuracy can bedetermined; however, the same point about accurate quantiﬁcation is valid. Such solutions thus highlight the generic diﬃculties that numerical methods must successfully tackle for more general structures. Reliable benchmarks,against which a solution obtained by such general-purpose numerical meth-ods can be veriﬁed, are needed to establish conﬁdence in the validity of thesecomputational methods in wider contexts where analysis becomes impossible.Exact or semi-analytic solutions are valuable elsewhere: in inverse scattering, exact solutions may pinpoint special eﬀects and distinguish between physi- cally real eﬀects and artefacts of the computational process. Moreover, manycanonical structures are of direct technological interest, particularly where ascattering process is dominated by that observed in a related canonical struc-ture. Mathematically, we solve a class of mixed boundary value problems and de- velop numerical formulations for computationally stable, rapidly convergingalgorithms of guaranteed accuracy. The potential problems and diﬀractionproblems are initially formulated as dual (or multiple) series equations, ordual (or multiple) integral equations. Central to the technique is the ideaof regularisation. The general concept of regularisation is well established in many areas of mathematics. In this context, its main feature is the transfor- mation of the badly behaved or singular part of the initial equations, describing a potential distribution or a diﬀraction process, to a well behaved set of equa- tions (technically, second-kind Fredholm equations). Physically, this processof semi-inversion corresponds to solving analytically some associated potentialproblem, and utilising that solution to determine the full wave scattering. The two volumes of this text are closely connected. Part I develops the theory of series equations and integral equations, and solves mixed bound-ary potential problems (mainly electrostatic ones) for structures with cavitiesand edges. The theory and structure of the dual equations that arise in thisprocess reﬂect new developments and reﬁnements since the major expositionof Sneddon [55]. In our uniﬁed approach, transformations connected with Abel’s integral equation are employed to invert analytically the singular part of the operator deﬁning the potential. Three-dimensional structures exam-ined include shells and cavities obtained by opening apertures in canonicallyshaped closed surfaces; thus a variety of spherical and spheroidal cavities andtoroidal and conical shells are considered. Although the main thrust of both volumes concerns three-dimensional eﬀects, some canonical two-dimensional structures, such as slotted elliptical cylinders and various ﬂat plates, are con-sidered. Also, to illustrate how regularisation transforms the standard integralequations of potential theory and beneﬁts subsequent numerical computa-tions, the method is applied to a noncanonical structure, the singly-slottedcylinder of arbitrary cross-section. Part II examines diﬀraction of acoustic and electromagnetic waves from ©200 1 CRC Press LLC similar classes of open structures with edges or cavities. The rigorous regu- larisation procedure relies on the techniques solutions developed in Part I toproduce eﬀective algorithms for the complete frequency range, quasi-static toquasi-optical. Physical interpretation of explicit mathematical solutions and relevant applications are provided. The two volumes aim to provide an account of some mathematical develop- ments over the last two decades that have greatly enlarged the set of soluble canonical problems of real physical and engineering signiﬁcance. They gather,perhaps for the ﬁrst time, a satisfactory mathematical description that accu-rately quantiﬁes the physically relevant scattering mechanisms in complex structures. Our selection is not exhaustive, but is chosen to illustrate the types of structures that may be analysed by these methods, and to provide aplatform for the further analysis of related structures. In developing a uniﬁed treatment of potential theory and diﬀraction, we have chosen a concrete, rather than an abstract or formal style of analysis. Thus, constructive methods and explicit solutions from which practical nu- merical algorithms can be implemented, are obtained from an intensive anduniﬁed study of series equations and integral equations. We hope this book will be useful to both new researchers and experienced specialists. Most of the necessary tools for the solution of series equationsand integral equations are developed in the text; allied material on special functions and functional analysis is collated in an appendix so that the book is accessible to as wide a readership as possible. It is addressed to mathemati-cians, physicists, and electrical engineers. The text is suitable for postgraduatecourses in diﬀraction and potential theory and related mathematical methods.It is also suitable for advanced-level undergraduates, particularly for projectmaterial. We wish to thank our partners and families for their support and encour- agement in writing this book. Their unfailing good humour and advice playeda key role in bringing the text to fruition. ©200 1 CRC Press LLC Chapter 1 Laplace’s Equation Laplace’s equation is one of the most important partial diﬀerential equations that arises in the application of mathematics to physical phenomena. It occursin diverse contexts, including electrostatics, magnetostatics, elasticity, grav-itation, steady-state heat conduction, incompressible ﬂuid ﬂow, and manyrelated areas described in, for example, [44] and [13]. Common to these disciplines is the notion of a potential ψ,which is a scalar function of spatial position. We will be particularly interested in the electro-static context, where the potential ψis constant on equipotential surfaces, and the associated electric ﬁeld vector− →Eis expressed via the gradient − →E=−∇ψ. (1. 1) This vector lies along the direction of most rapid decrease of ψ.Gauss’ law states that the divergence of the electric ﬁeld is proportional to charge density ρat each point in space, ∇.− →E= 4πρ. (1. 2) The proportionality factor in Equation (1. 2) depends upon the choice of units. We employ Gaussian units [20] throughout; if Syst` eme International (SI) units are employed, the right-hand side of (1. 2) is divided by 4 πε 0where εodenotes free space permittivity. (To convert capacitances from Gaussian to SI units, multiply by 4 πε0). From (1. 1) and (1. 2), Poisson’s equation follows, ∇.(∇ψ) =∇2ψ=−4πρ. (1. 3) This equation describes how the potential is determined by the charge distri-bution in some region of space. Now consider an electrostatic ﬁeld with associated potential ψ.If a perfectly conducting surface Sis immersed in this ﬁeld, a charge distribution ρ iis induced on the surface; it has an associated potential ψisatisfying (1. 3). The total potential Ψ = ψ+ψiis constant on S(an equipotential surface), the total electric ﬁeld −∇Ψ is normal to S(at each point), and because there are no charges except on S, the total potential satisﬁes Laplace’s equation, ∇2ψ= 0, (1. 4) ©200 1 CRC Press LLC at every point of space except on S. In order to obtain a unique solution that is physically relevant, this partial diﬀerential equation must be complemented by appropriate boundary condi- tions; for example, the potential on one or more metallic conductors might be speciﬁed to be of unit value, and Laplace’s equation is to be solved in the region excluding the conductors, but subject to this speciﬁcation on the con-ductor surface. If one of the conductors encloses a (ﬁnite) region of interest,such boundary conditions may be suﬃcient to specify the required solutionuniquely; however, in unbounded regions, some additional speciﬁcation of thebehaviour of the potential at inﬁnity is required. Moreover, the presence of sharp edges on the bounding conducting surfaces may require that additional constraints, equivalent to the ﬁniteness of energy, be imposed to ensure thata physically relevant solution is uniquely deﬁned by Laplace’s equation. In this book we shall be interested in analytic and semi-analytic methods for solving Laplace’s equation with appropriate boundary and other conditions. To make substantive progress, we shall consider orthogonal coordinate systems in which Laplace’s equation is separable (i.e., it can be solved by the methodof separation of variables), and the conductors occupy part or whole of acoordinate surface in these systems. Laplace’s equation can be solved by the method of separation of variables only when the boundary conditions are enforced on a complete coordinate surface (e.g., the surface of a sphere in the spherical coordinate system). As indicated in the preface, it is important to emphasize that the methods de-scribed in this book apply to a much wider class of surfaces, where the bound-ary conditions (describing, say, the electrostatic potential of a conductor) areprescribed on only part of a coordinate surface in the following way. Letu 1,u2,andu3be a system of coordinates in which the three sets of coordinate surfaces,u1= constant, u2= constant, and u3= constant, are mutually or- thogonal. We shall consider portions of a coordinate surface typically speciﬁedby u 1= constant, a≤u2≤b (1. 5) whereaandbare ﬁxed. For example, a spherical cap of radius aand sub- tending an angle θo(at the centre of the appropriate sphere) may be speciﬁed in the spherical coordinate system ( r,θ,φ ) by r=a,0≤θ≤θ0,0≤φ≤2π. (1. 6) The determination of the electrostatic potential surrounding the cap can be posed as a mixed boundary value problem , and can be solved by the analytic methods of this book, despite its insolubility by the method of separation of variables. Although the type of surface speciﬁed by (1. 5) is somewhat restricted, it includes many cases not merely of mathematical interest, but of substantive ©200 1 CRC Press LLC physicalandtechnologicali nterestaswell ;theclassofsurfacesforwhichana- lyticsolution stothepote ntialtheor yproblem(ofsolvingLaplace’sequation) canbefoundisthu sconsiderablyenlarged,beyondth ewell-establishe dclass ofsolutionsobtaine dbyseparationofvariables(see,forexampl e[54]).Since itwillbece ntraltolate rdevelopments ,Section s1.1and1.2brieﬂ ydescribe theformofLaplace’sequationi nsomeo ftheseorthogonalc oordinatesys- tems,andthesolution sgenerate dbyth eclassicalmeth odofseparationof variables. Theformulationofpote ntialtheor yforstructureswithedge sisexpounded inSection1.3.Fortheclas sofsurfacesdescribe dabove,dual(ormultiple) seriesequationsarisenaturally,asd odual(ormultiple )integralequations. Variousmethodsforsolvingsuchdualseriesequation saredescribedinSec- tion1.4,includin gtheAbelintegraltransformmethodthati sthekeytool employe dthroughou tthistext.Itexploitsfeature sofAbel’sintegralequation (describe dinSection1.5)andAbel-typei ntegralrepresentation sofLegendre polynomials ,Jacob ipolynomials,andrelate dhypergeometricfunctions(de- scribedinSection1.6).I ntheﬁnalSection(1.7),th eequivalenc eofthedual seriesapproachan dthemoreusualintegralequationapproach(employing single-ordouble-layersurfacedensities )topotentia ltheoryisdemonstrated. 1.1Laplace’sequationi ncurvilinearcoordinates ThestudyofLaplace’ sequationinvariousc oordinatesystemshasalong history,generating,amongstotheras pects,manyofth especialfunction sof appliedmathematicsan dphysic s(Besselfunctions ,Legendr efunctions ,etc.). Inthissectionwegathermaterialofareferencenature;foragreate rdepth ofdetail ,werefe rtheintereste dreade rtooneofthenumeroustextswritten onthesetopics ,suchas[44],[32]or[74]. HereweconsiderLaplace’sequationinthos ecoordinatesystemsthatwill beofconcreteinteres tlate rinthi sbook;i nthesesystemsth emethodof separationofvariablesisapplicable.Le tu1,u2,andu3beasystemofc oor- dinatesinwhichthec oordinatesurface su1=constant, u2=constant,and u3=constantaremutuall yorthogonal(i.e. ,intersec torthogonally).Fixa point(u1,u2,u3)andconsiderth eeleme ntaryparallelepipe dforme dalon gthe coordinatesurfaces ,asshowninFigure1.1. Thus O, A, B, and C have coordinates ( u1,u2,u3),(u1+du1,u2,u3),(u1,u2+ du2,u3),and (u1,u2,u3+du3),respectively .The length dsof the diagonal line segment connecting ( u1,u2,u3) and (u1+du1,u2+du2,u3+du3) is given by ds2=h2 1du21+h22du2+h33du23(1. 7) whereh1,h2,andh3are the metric coeﬃcients (or Lam´ e coeﬃcients, in recog- ©200 1 CRC Press LLC Figure 1.1 The elementary parallelepiped. nition of the transformation of the Laplacian to general orthogonal coordinates ﬁrst eﬀected in [35]). In terms of the Lam´ e coeﬃcients, the lengths of the elementary paral- lelepiped edges equal h1du1,h2du2,andh3du3,respectively, so that its volume ish1h2h3du1du2du3.These coeﬃcients depend, in general, upon the coordi- natesu1,u2,u3and can be calculated explicitly from the functional relation- ship between rectangular and curvilinear coordinates, x=x(u1,u2,u3), y=y(u1,u2,u3), z=z(u1,u2,u3). (1. 8) It is useful to state the relationship between rectangular and curvilinear com- ponents of any vector− →F.Designate by− →ix,− →i y,− →izthe unit rectangular (Carte- sian) coordinate vectors, and by− →i1,− →i 2,− →i3the unit coordinate vectors in the orthogonal curvilinear coordinate system; the unit vectors are deﬁned by the relation (with− →r=x− →ix+y− →iy+z− →iz): − →ii=1 hi/parenleftbigg∂x ∂ui− →ix+∂y ∂ui− →iy+∂z ∂ui− →iz/parenrightbigg =∂− →r ∂ui//vextendsingle/vextendsingle/vextendsingle/vextendsingle∂− →r ∂ui/vextendsingle/vextendsingle/vextendsingle/vextendsingle(1. 9) wherei= 1,2,3,and are mutually orthogonal. Then − →F=F x− →ix+Fy− →iy+Fz− →iz=F1− →i1+F2− →i2+F3− →i3. (1. 10) Taking inner products yields the following relations: F1=Fx(− →ix,− →i1) +Fy(− →iy,− →i1) +Fz(− →iz,− →i1) F2=Fx(− →ix,− →i2) +Fy(− →iy,− →i2) +Fz(− →iz,− →i2) (1. 11) F3=Fx(− →ix,− →i3) +Fy(− →iy,− →i3) +Fz(− →iz,− →i3) ©200 1 CRC Press LLCh du 33G FH CI A BOh du2 211h du Thediﬀerential softherectangularcoordinate sarelinearfunction softhe curvilinearcoordinates: dx=∂x ∂u1du1+∂x ∂u2du2+∂x ∂u3du3, dy=∂y ∂u1du1+∂y ∂u2du2+∂y ∂u3du3, (1.12) dz=∂z ∂u1du1+∂z ∂u2du2+∂z ∂u3du3. Comparingtheexpressionforelementarylength ds2=dx2+dy2+dz2with (1.7),andusingorthogonalityofth ecoordinatebasisvectors ,weobtain h2 1du21+h22du22+h23du23=dx2+dy2+dz2;(1.13) substituting(1.12)i nto(1.13)an dequatinglikecoeﬃcientssh owsthat h2 i=/parenleftbigg∂x ∂ui/parenrightbigg2 +/parenleftbigg∂y ∂ui/parenrightbigg2 +/parenleftbigg∂z ∂ui/parenrightbigg2 (i=1,2,3). (1.14) Letψ=ψ(u1,u2,u3)beascalarfunctiondependentuponspatialposition, andlet−→A=−→A(u1,u2,u3)beavectorfunctionofposition ,thethreecom po- nentsofwhi chwil lbedenoted A1=A1(u1,u2,u3),A2=A2(u1,u2,u3),and A3=A3(u1,u2,u3).Wewis htoﬁn dthecoordinateexpressionforth egradi- entofthescalar ψ(gradψ)inthissystem,aswellasthedivergence(div−→A) andcirculationorcurl(curl−→A)ofth evector−→A. Itfoll owsfromFigure1. 1thattheﬁrstcomponentofthegradie ntis (gradψ)1= lim du1→0ψ(u1+du1,u2,u3)−ψ(u1,u2,u3) h1du1=1 h1∂ψ ∂u1.(1. 15) Analogously, the other two components are (gradψ)2=1 h2∂ψ ∂u2,(gradψ)3=1 h3∂ψ ∂u3. (1. 16) To determine the divergence, let us calculate the total ﬂux /integraldisplay S− →A.− →nds of the vector− →Athrough the surface Sof the elementary parallelepiped, the ﬂux being calculated in the direction of the external unit normal− →n.The ﬂux through the surface OBHC is A1h2h3du2du3,whereas the ﬂux through surface AFGI is A1h2h3du2du3+∂ ∂u1(A1h2h3)du1du2du3, ©200 1 CRC Press LLC so the net ﬂux through these two surfaces is ∂ ∂u1(A1h2h3)du1du2du3. The net ﬂux through the remaining two opposing pairs of surfaces is ∂ ∂u2(A2h3h1)du1du2du3and∂ ∂u3(A3h1h2)du1du2du3. Thus the total ﬂux through the complete parallelepiped surface is /integraldisplay S− →A.− →nds=/bracketleftbigg∂ ∂u1(A1h2h3) +∂ ∂u2(A2h3h1) +∂ ∂u3(A3h1h2)/bracketrightbigg du1du2du3. According to the Gauss-Ostrogradsky theorem [74], [32] /integraldisplay S− →A.− →nds=/integraldisplay Vdiv− →AdV whereVis the volume enclosed by S. A comparison of the last two formulae shows that in curvilinear coordinates the divergence of− →Ais (also denoted ∇.− →A), div− →A=1 h1h2h3/bracketleftbigg∂ ∂u1(A1h2h3) +∂ ∂u2(A2h3h1) +∂ ∂u3(A3h1h2)/bracketrightbigg .(1. 17) To derive the circulation (curl− →A) of the vector− →A, consider the contour OBHC, which is denoted L. Observing that /integraldisplayB 0− →A.− →dl=A2h2du2, /integraldisplayC H− →A.− →dl=−A2h2du2−∂ ∂u3(A2h2du2)du3, /integraldisplayH B− →A.− →dl=A3h3du3+∂ ∂u2(A3h3du3)du2, /integraldisplayO C− →A.− →dl=−A3h3du3, the circulation along this contour Lis /contintegraldisplay L− →A.− →dl=∂ ∂u2(A3h3du3)du2−∂ ∂u3(A2h2du2)du3. According to Stokes’ theorem [74], [32] /contintegraldisplay L− →A.− →dl=/integraldisplay Scurl− →A.− →nds ©200 1 CRC Press LLC whereSis the surface bounded by L, with the normal− →ndeﬁned above. A comparison of the last two formulae shows that the circulation curl− →A≡ ∇×− →A has ﬁrst component (curl− →A)1=1 h2h3/bracketleftbigg∂ ∂u2(h3A3)−∂ ∂u3(h2A2)/bracketrightbigg . (1. 18) Considering the contours OCIA and OAFB, the other two components are (curl− →A)2=1 h3h1/bracketleftbigg∂ ∂u3(h1A1)−∂ ∂u1(h3A3)/bracketrightbigg , (1. 19) (curl− →A)3=1 h1h2/bracketleftbigg∂ ∂u1(h2A2)−∂ ∂u2(h1A1)/bracketrightbigg . (1. 20) The Laplacian can now be stated in curvilinear coordinate form, combining (1. 15), (1. 16), and (1. 17) with the deﬁnition /triangleψ=∇2ψ= div(grad ψ) (1. 21) to obtain ∇2ψ=1 h1h2h3/bracketleftbigg∂ ∂u1(h2h3 h1∂ψ ∂u1) +∂ ∂u2/parenleftbiggh3h1 h2∂ψ ∂u2/parenrightbigg +∂ ∂u3/parenleftbiggh1h2 h3∂ψ ∂u3/parenrightbigg/bracketrightbigg . (1. 22) Let us gather the explicit form of the metric coeﬃcients, the volume ele- ment, and the Laplacian in the various coordinates systems of interest in this book. 1.1.1 Cartesian coordinates The range of the coordinates is −∞<x< ∞,−∞<y< ∞,−∞<z< ∞. The metric coeﬃcients are hx=hy=hz= 1,and the volume element is dV=dxdydz. The forms of the Laplacian and gradient are, respectively, /triangleψ=∂2ψ ∂x2+∂2ψ ∂y2+∂2ψ ∂z2= 0, (1. 23) ∇ψ=− →ix∂ψ ∂x+− →iy∂ψ ∂y+− →iz∂ψ ∂z. (1. 24) The coordinates surfaces ( x,y, orz= constant) are planes. ©200 1 CRC Press LLC 1.1.2 Cylindrical polar coordinates In terms of Cartesian coordinates, the cylindrical coordinates are x=ρcosφ, y=ρsinφ, z=z, and the range of the coordinates is 0 ≤ρ <∞,0≤φ≤2π,−∞< z < ∞. The metric coeﬃcients are hρ= 1, hφ=ρ, h z= 1, and the volume element is dV=ρdρdφdz . The forms of the Laplacian and gradient are, respectively, /triangleψ=1 ρ∂ ∂ρ/parenleftbigg ρ∂ψ ∂ρ/parenrightbigg +1 ρ2∂2ψ ∂φ2+∂2ψ ∂z2, (1. 25) ∇ψ=− →iρ∂ψ ∂ρ+− →iφ1 ρ∂ψ ∂φ+− →iz∂ψ ∂z. (1. 26) The coordinates surfaces are cylinders ( ρ= constant), planes through the z-axis (φ= constant), or planes perpendicular to the z-axis (z= constant). 1.1.3 Spherical polar coordinates In terms of Cartesian coordinates, the spherical coordinates are x=rsinθcosφ, y=rsinθsinφ, z=rcosθ, and the range of the coordinates is 0 ≤r <∞,0≤θ≤π,0≤φ≤2π.The metric coeﬃcients are hr= 1, hθ=r, h φ=rsinθ. The volume element is dV=r2sinθdrdθdφ and the forms of the Laplacian and gradient are, respectively, /triangleψ=1 r2∂ ∂r/parenleftbigg r2∂ψ ∂r/parenrightbigg +1 r2sinθ∂ ∂θ/parenleftbigg sinθ∂ψ ∂θ/parenrightbigg +1 r2sin2θ∂2ψ ∂φ2,(1. 27) ∇ψ=− →ir∂ψ ∂r+− →iθ1 r∂ψ ∂θ+− →iφ1 rsinθ∂ψ ∂φ. (1. 28) The coordinates surfaces are spheres ( r= constant), right circular cones ( θ= constant), or azimuthal planes containing the z-axis (φ= constant). ©200 1 CRC Press LLC 1.1.4 Prolate spheroidal coordinates There are two commonly used systems of spheroidal coordinates employing coordinates denoted ( ξ,η,ϕ ) and (α,β,ϕ ),respectively. In terms of Cartesian coordinates, the ﬁrst representation is x=d 2/radicalbig (1−η2)(ξ2−1) cosϕ, y =d 2/radicalbig (1−η2)(ξ2−1) sinϕ, z =d 2ηξ, where the parameter dwill be identiﬁed as the interfocal distance; the range of coordinates is 1 ≤ξ<∞,−1≤η≤1,0≤φ<2π. The coordinate surface ξ= constant >1 is a prolate spheroid with foci at the points ( x,y,z ) = (0,0,±d 2),with major semi-axis b=d 2ξ, and minor semi-axisa=d 2/parenleftbig ξ2−1/parenrightbig1 2, x2+y2 (ξ2−1)+z2 ξ2=/parenleftbiggd 2/parenrightbigg2 ; the degenerate surface ξ= 1 is the straight line segment |z| ≤d 2. The coor- dinate surface |η|= constant <1 is a hyperboloid of revolution of two sheets with an asymptotic cone whose generating line passes through the origin and is inclined at an angle β= cos−1(η) to thez−axis, z2 η2−x2+y2 (1−η2)=/parenleftbiggd 2/parenrightbigg2 ; the degenerate surface |η|= 1 is that part of the z−axis for which |z|>1 2d. The surface ϕ= constant is a half-plane containing the z−axis and forming angleϕwith thex,z−plane. In the limit when the interfocal distance approaches zero and ξtends to inﬁnity, the prolate spheroidal system ( ξ,η,ϕ ) reduces to the spherical system (r,θ,φ sphere ) by making the identiﬁcation d 2ξ=r, η= cosθ, ϕ≡φsphere in such a way that the productd 2ξremains ﬁnite as d→0,ξ→ ∞ . The second representation ( α,β,ϕ ) of prolate spheroidal coordinates is ob- tained by setting ξ= coshαandη= cosβso that in terms of Cartesian coordinates x=d 2sinhαsinβcosϕ, y =d 2sinhαsinβsinϕ, z =d 2coshαcosβ. The range of coordinates is 0 ≤α <∞,0≤β≤π,0≤φ < 2π. Both representations are used equally in this book. ©200 1 CRC Press LLC The metric coeﬃcients are, respectively, hξ=d 2/radicalBigg ξ2−η2 ξ2−1, hη=d 2/radicalBigg ξ2−η2 1−η2, hφ=d 2/radicalbig (ξ2−1) (1−η2) and hα=hβ=d 2/radicalBig sinh2α+ sin2β, h φ=d 2sinhαsinβ; the volume element is dV=/parenleftbiggd 2/parenrightbigg3/parenleftbig ξ2−η2/parenrightbig dξdηdφ =/parenleftbiggd 2/parenrightbigg3/parenleftbig sinh2α+ sin2β/parenrightbig sinhαsinβdαdβdφ. The forms of the Laplacian and gradient are, respectively, /parenleftbiggd 2/parenrightbigg2 /triangleψ=1 (ξ2−η2)/braceleftbigg∂ ∂ξ/parenleftbigg/parenleftbig ξ2−1/parenrightbig∂ψ ∂ξ/parenrightbigg +∂ ∂η/parenleftbigg/parenleftbig 1−η2/parenrightbig∂ψ ∂η/parenrightbigg/bracerightbigg +1 (ξ2−1) (1−η2)∂2ψ ∂φ2,(1. 29) /parenleftbiggd 2/parenrightbigg ∇ψ=− →iξ/radicalBigg ξ2−1 ξ2−η2∂ψ ∂ξ+− →iη/radicalBigg 1−η2 ξ2−η2∂ψ ∂η +− →iφ/bracketleftbig/parenleftbig ξ2−1/parenrightbig/parenleftbig 1−η2/parenrightbig/bracketrightbig−1 2∂ψ ∂φ,(1. 30) and /parenleftbiggd 2/parenrightbigg2/parenleftbig sinh2α+ sin2β/parenrightbig /triangleψ =1 sinhα∂ ∂α/parenleftbigg sinhα∂ψ ∂α/parenrightbigg +1 sinβ∂ ∂β/parenleftbigg sinβ∂ψ ∂β/parenrightbigg +/parenleftbigg1 sinh2α+1 sin2β/parenrightbigg∂2ψ ∂φ2,(1. 31) /parenleftbiggd 2/parenrightbigg ∇ψ=1/radicalbig sinh2α+ sin2β/braceleftbigg− →iα∂ψ ∂α+− →iβ∂ψ ∂β/bracerightbigg +− →iφ1 sinhαsinβ∂ψ ∂φ. (1. 32) ©200 1 CRC Press LLC 1.1.5 Oblate spheroidal coordinates As with the prolate system, there are two commonly used systems of oblate spheroidal coordinates employing coordinates denoted ( ξ,η,ϕ ) and (α,β,ϕ ), respectively. In terms of Cartesian coordinates, the ﬁrst representation is x=d 2/radicalbig (1−η2) (ξ2+ 1) cosφ, y=d 2/radicalbig (1−η2) (ξ2+ 1) sinφ, z=d 2ηξ where the parameter dwill be identiﬁed as interfocal distance; the range of the coordinates is 0 ≤ξ <∞,−1≤η≤1,0≤φ < 2π. The coordinate surfaceξ= constant is an oblate spheroid with foci at the points ( x,y,z ) = ±/parenleftbigd 2,d 2,0/parenrightbig , x2+y2 (ξ2+ 1)+z2 ξ2=/parenleftbiggd 2/parenrightbigg2 ; the degenerate surface ξ= 0 is the disk x2+y2≤/parenleftbigd 2/parenrightbig2in the plane z= 0. The coordinate surface η= constant is a one-sheeted hyperboloid of revolution, with an asymptotic cone whose generating line passes through the origin and is inclined at the angle β= cos−1(η) to thez−axis, x2+y2 (1−η2)−z2 η2=/parenleftbiggd 2/parenrightbigg2 . The coordinate surface φ= constant is a half-plane containing the z-axis. The second representation ( α,β,ϕ ) of oblate spheroidal coordinates is ob- tained by setting ξ= sinhαandη= cosβso that in terms of Cartesian coordinates x=d 2coshαsinβcosϕ, y =d 2coshαsinβsinϕ, z =d 2sinhαcosβ, where the range of coordinates is 0 ≤α<∞,0≤β≤π,0≤φ<2π. The metric coeﬃcients are, respectively, hξ=d 2/radicalBigg ξ2+η2 ξ2+ 1, hη=d 2/radicalBigg ξ2+η2 1−η2, hφ=d 2/radicalbig (ξ2+ 1) (1 −η2), and hα=hβ=d 2/radicalBig cosh2α−sin2β, h φ=d 2coshαsinβ. The forms of the Laplacian and gradient are, respectively, /parenleftbiggd 2/parenrightbigg2 /triangleψ=1 (ξ2+η2)/braceleftbigg∂ ∂ξ/parenleftbigg/parenleftbig ξ2+ 1/parenrightbig∂ψ ∂ξ/parenrightbigg +∂ ∂η/parenleftbigg/parenleftbig 1−η2/parenrightbig∂ψ ∂η/parenrightbigg/bracerightbigg +1 (ξ2+ 1) (1 −η2)∂2ψ ∂φ2,(1. 33) ©200 1 CRC Press LLC /parenleftbiggd 2/parenrightbigg ∇ψ=− →iξ/radicalBigg ξ2+ 1 ξ2+η2∂ψ ∂ξ+− →iη/radicalBigg 1−η2 ξ2+η2∂ψ ∂η +− →iφ1/radicalbig (ξ2+ 1) (1 −η2)∂ψ ∂φ,(1. 34) and /parenleftbiggd 2/parenrightbigg2/parenleftbig cosh2α−sin2β/parenrightbig /triangleψ =1 coshα∂ ∂α/parenleftbigg coshα∂ψ ∂α/parenrightbigg +1 sinβ∂ ∂β/parenleftbigg sinβ∂ψ ∂β/parenrightbigg +/parenleftbigg1 sin2β−1 cosh2α/parenrightbigg∂2ψ ∂φ2,(1. 35) /parenleftbiggd 2/parenrightbigg ∇ψ=1/radicalbig cosh2α−sin2β/braceleftbigg− →iα∂ψ ∂α+− →iβ∂ψ ∂β/bracerightbigg +− →iφ1 coshαsinβ∂ψ ∂φ. (1. 36) 1.1.6 Elliptic cylinder coordinates In terms of Cartesian coordinates, the elliptic cylinder coordinates are x=d 2coshαcosβ, y =d 2sinhαsinβ, z=z, where the range of the coordinates is −∞<α< ∞,0≤β≤π,−∞<z< ∞. The metric coeﬃcients are hα=hβ=d 2/radicalBig cosh2α−cos2β,hz= 1, and the volume element is dV=/parenleftbigd 2/parenrightbig3/parenleftbig cosh2α−cos2β/parenrightbig .The forms of the Laplacian and gradient are, respectively, /triangleψ=1 /parenleftbigd 2/parenrightbig2/parenleftbig cosh2α−cos2β/parenrightbig/braceleftbigg∂2ψ ∂α2+∂2ψ ∂β2/bracerightbigg +∂2ψ ∂z2, (1. 37) ∇ψ=1 /parenleftbigd 2/parenrightbig/parenleftbig cosh2α−cos2β/parenrightbig1 2/braceleftbigg− →iα∂ψ ∂α+− →iβ∂ψ ∂β/bracerightbigg +− →iz∂ψ ∂z. An alternative representation employs ξ= coshα,η= cosβ,so that x=d 2ξη, y =d 2/radicalbig (ξ2−1) (1−η2), z=z, ©200 1 CRC Press LLC where the range of the coordinates is 1 ≤ξ <∞,−1≤η≤1,−∞<z < ∞. The metric coeﬃcients are hξ=d 2/radicalBigg ξ2−η2 ξ2−1, hη=d 2/radicalBigg ξ2−η2 1−η2, hz= 1. The volume element is dV=/parenleftbigd 2/parenrightbig3/parenleftbig ξ2−η2/parenrightbig/braceleftbig/parenleftbig ξ2−1/parenrightbig/parenleftbig 1−η2/parenrightbig/bracerightbig−1 2dξdηdz . The forms of the Laplacian and gradient are, respectively, /triangleψ=/radicalbig ξ2−1 /parenleftbigd 2/parenrightbig2(ξ2−η2)∂ ∂ξ/parenleftbigg/radicalbig ξ2−1∂ψ ∂ξ/parenrightbigg + /radicalbig 1−η2 /parenleftbigd 2/parenrightbig2(ξ2−η2)∂ ∂η/parenleftbigg/radicalbig 1−η2∂ψ ∂η/parenrightbigg +∂2ψ ∂z2(1. 38) ∇ψ=− →iξ/parenleftbiggd 2/parenrightbigg−1/radicalBigg ξ2−1 ξ2−η2∂ψ ∂ξ+− →iη/parenleftbiggd 2/parenrightbigg−1/radicalBigg 1−η2 ξ2−η2∂ψ ∂η+− →iz∂ψ ∂z(1. 39) The coordinate surfaces are confocal elliptic cylinders with semi-focal distance d 2(whenξorαis constant) or confocal, one-sheeted hyperbolic cylinders (when ηorβis constant), or planes perpendicular to the z-axis (z= constant). 1.1.7 Toroidal coordinates In terms of Cartesian coordinates, the toroidal coordinates employ a scale factorc>0 and x=csinhαcosφ coshα−cosβ, y=csinhαsinφ coshα−cosβ, z=csinβ coshα−cosβ, where the range of the coordinates is 0 ≤α<∞,−π≤β≤π,−π≤φ≤π. The metric coeﬃcients are hα=hβ=c coshα−cosβ, hφ=csinhα coshα−cosβ, and the volume element is dV=c3sinhα(coshα−cosβ)−3dαdβdφ. The form of the Laplacian and gradient can be expressed as hαhβhφ/triangleψ=∂ ∂α/parenleftbigg hφ∂ψ ∂α/parenrightbigg +∂ ∂β/parenleftbigg hφ∂ψ ∂β/parenrightbigg +1 (coshα−cosβ) sinhα∂2ψ ∂φ2, (1. 40) ∇ψ=− →iαc−1(coshα−cosβ)∂ψ ∂α+− →iβc−1(coshα−cosβ)∂ψ ∂β +− →izc−1(coshα−cosβ) sinhα∂ψ ∂φ.(1. 41) ©200 1 CRC Press LLC Thec oordinatesurface scorrespondingtoconstant αaretori(withminorra- diusr=c/sinhαandm ajorradiu sR=ccothα,thetoriare/parenleftBig/radicalbig x2+y2−R/parenrightBig2 + z2=r2);forconstant β,thecoordinatesurface saresphereso fradiusa= c/sinβandcentreonthe z-axisat(x,y,z )=(0,0,b),wher eb=ccotβ;the coordinatesurfacesofconstant φareazimuthalplane scontainingth ez-axis. (SeeFigur e5.1.) 1.2 Solutions of Laplace’s equation: separation of vari- ables In this section we describe the solutions to Laplace’s equation generated by the classical method of separation of variables. A knowledge of these solutions is essential for the approach to the solution of mixed boundary value problems described in the next section, because it depends upon the formulation of anappropriate set of dual series equations with special function kernels. 1.2.1 Cartesian coordinates We seek a solution to Laplace’s equation in the form ψ(x,y,z ) =X(x)Y(y)Z(z). (1. 42) Substitution in Equation (1. 23) transforms it to 1 Xd2X dx2+1 Yd2Y dy2+1 Zd2Z dz2= 0. (1. 43) Each term in this equation is a function of only one independent variable, sothere are constants (“separation constants”) νandµsuch that 1 Xd2X dx2=−ν2⇒X/prime/prime+ν2X= 0, (1. 44) 1 Yd2Y dy2=−µ2⇒Y/prime/prime+ν2Y= 0, (1. 45) and hence 1 Zd2Z dz2−/parenleftbig ν2+µ2/parenrightbig = 0⇒Z/prime/prime−/parenleftbig ν2+µ2/parenrightbig Z= 0. (1. 46) Thus the original equation involving partial derivatives has been reduced tothree ordinary diﬀerential equations. The process just described is the classical process of separation of variables and leads to inﬁnitely many solutions of the form (1. 42), depending on the ©200 1 CRC Press LLC parameters νandµ,which can take real or complex values. The solution of Equations (1. 44)–(1. 46) can be expressed in terms of elementary functions of form Xν(x) =Aνcosνx+Bνsinνx, (1. 47) Yµ(y) =Cµcosµy+Dµsinµy, (1. 48) and Zν,µ(z) =Eν,µe−√ ν2+µ2z+Fν,µe+√ ν2+µ2z, (1. 49) whereAν,Bν,Cµ,Dµ,Eν,µ,andFν,µare constants. The required solution of the given physical problem is obtained by linear superposition of the particular solutions (1. 42) formed from (1. 47)–(1. 49), of the form /summationdisplay ν,µXν(x)Yµ(y)Zν,µ(z) or/integraldisplay/integraldisplay Xν(x)Yµ(y)Zν,µ(z)dνdµ, where the speciﬁc conditions of the problem dictate the range of parameters ν,µused in the summation or integration as appropriate. 1.2.2 Cylindrical polar coordinates Applying the method of separation of variables, the Laplace Equation (1. 25)has particular solutions of the form ψ(ρ,φ,z ) =R(ρ)Φ(φ)Z(z), (1. 50) where 1 ρd dρ(ρdR dρ) +/parenleftbigg λ2−µ2 ρ2/parenrightbigg R= 0, (1. 51) d2Φ dφ2+µ2Φ = 0, (1. 52) d2Z dz2−λ2Z= 0, (1. 53) andλandµare the “separation constants.” The solutions of the latter two equations are the same as those considered above in (1. 44) and (1. 46): Φµ(φ) =Aµcos(µφ) +Bµsin(µφ), (1. 54) Zλ(z) =Cλe−λz+Dλe+λz. (1. 55) Equation (1. 51) cannot be expressed in terms of elementary functions; rescalin gu=λρ,weobtai nBessel’ sdiﬀere ntialequatio n(seeAppendixB.5), ud du(udR du) + (u2−µ2)R= 0. (1. 56) ©200 1 CRC Press LLC Its solutions are linear combinations of Bessel functions, Rλ,µ(ρ) =Eλ,µJµ(λρ) +Fλ,µYµ(λρ), (1. 57) whereJµ(λρ) andYµ(λρ) are the Bessel functions of order µ,of ﬁrst and second kind, respectively. 1.2.3 Spherical polar coordinates In spherical polars, the Laplace Equation (1. 27) has separated solutions ψ(r,θ,φ ) =R(r)Θ(θ)Φ(φ) (1. 58) where 1 r2d dr(r2dR dr)−ν(ν+ 1) r2R= 0, (1. 59) 1 sinθd dθ/parenleftbigg sinθdΘ dθ/parenrightbigg +/bracketleftbigg ν(ν+ 1)−µ2 sin2θ/bracketrightbigg Θ = 0, (1. 60) d2Φ dφ2+µ2Φ = 0, (1. 61) andµ,νare the most conveniently chosen forms of the separation constants. The solutions of these equations are R(r) =Aνrν+Bνr−ν−1, (1. 62) Θ(θ) =Cν,µPµ ν(cosθ) +Dν,µQµ ν(cosθ), (1. 63) Φ(φ) =Eµcosµφ+Fµsinµφ, (1. 64) wherePµ ν(cosθ) andQµ ν(cosθ) are the associated Legendre functions (see Ap- pendix B.4) of the ﬁrst and second kind, respectively. When boundary condi- tions are applied on spherical coordinate surfaces, no boundaries of which lie along the planes φ= constant, enforcement of continuity and of periodicity upon Φ requires that µbe zero or a positive integer, i.e., µ=m(m= 0,1,2...). The Legendre functions Pm ν(cosθ) are ﬁnite over the range 0 ≤θ≤πonly whenνis an integer n,equal tom,or larger. These requirements, of period- icity of the solution over the range 0 ≤θ≤π,and of its ﬁniteness, restrict the separation constants so that the particular solutions of Laplace’s equation inspherical coordinates are linear combinations of r nY(e) mn, rnY(o) mn, r−n−1Y(e) mn,andr−n−1Y(o) mn, where Y(e) mn= cos(mφ)Pm n(cosθ) andY(o) mn= sin(mφ)Pm n(cosθ) (1. 65) are the “spherical harmonics.” Those harmonics with m= 0 are zonal har- monics (since these functions depend only on θ, the nodal lines divide the sphere into zones), those with m=nare sectoral harmonics (since these functions depend only on φ,the nodal lines divide the sphere into sectors), and the rest, for 0 <m<n, are known as tesseral harmonics. Their properties aredescri bedinthereference sinAppendixB. ©200 1 CRC Press LLC 1.2.4 Prolate spheroidal coordinates The separated solutions of Laplace’s equation in prolate spheroidal coordi- nates (1. 29) are ψ(ξ,η,φ ) =X(ξ)H(η)Φ(φ), where d dξ/bracketleftbigg (ξ2−1)dX dξ/bracketrightbigg −/bracketleftbigg n(n+ 1) +m2 ξ2−1/bracketrightbigg X= 0, (1. 66) d dη/bracketleftbigg (1−η2)dH dη/bracketrightbigg +/bracketleftbigg n(n+ 1)−m2 1−η2/bracketrightbigg H= 0, (1. 67) d2Φ dφ2+m2Φ = 0. (1. 68) The separation constants are nandm. Admissible solutions of the third equation, with periodic boundary conditions on Φ ,are Φm(φ) =Emcos(mφ) +Dmsin(mφ), (1. 69) wheremis zero or a positive integer. The ﬁrst and second equations have as solutions the associated Legendre functions Pm nandQm nof the ﬁrst and second kind. For the second equation, if η∈[−1,1], the only ﬁnite solutions (atη=±1) forHmust be proportional to the Legendre function of the ﬁrst kind,Pm n(η),wherenis zero or a positive integer; if this restriction is removed H(η) =Cm nPm n(η) +Dm nQm n(η). (1. 70) The maximum range of the variable ξis [1,∞).For most values of nandm there is no solution to (1. 66) which is ﬁnite over the whole of this interval, so we use whatever linear combination of Pm n(ξ) andQm n(ξ) that is ﬁnite inside the boundaries of the problem, X(ξ) =Am nPm n(ξ) +Bm nQmn(ξ). (1. 71) In this way, the partial solution of Laplace’s equation ψnm(ξ,η,φ ) is the product of (1. 69)–(1. 71). In the alternative representation of Laplace’s Equation (1. 31), the sepa- rated solutions take the form ψ(α,β,φ ) =A(α)B(β)Φ(φ), where Φ satisﬁes (1. 68); Asatisﬁes 1 sinhαd dα/parenleftbigg sinhαdA dα/parenrightbigg −/bracketleftbigg n(n+ 1) +m2 sinh2α/bracketrightbigg A= 0, (1. 72) so that it is a linear combination of Pm n(coshα) andQm n(coshα); andBsat- isﬁes 1 sinβd dβ/parenleftbigg sinβdB dβ/parenrightbigg +/bracketleftbigg n(n+ 1)−m2 sin2β/bracketrightbigg B= 0, (1. 73) so that it is a linear combination of Pm n(cosβ) andQm n(cosβ). ©200 1 CRC Press LLC 1.2.5 Oblate spheroidal coordinates The separated equations for the θ- andη- coordinates are the same as for prolate spheroids, generating solutions sin mθ,cosmθandPm n(η),wherem andnare positive integers (or zero). The equation for the ξ- coordinate has solutionsPm n(iξ) andQm n(iξ).Thus, the partial solutions of Laplace’s equation in this system have the form φnm(ξ,η,θ ) = [Am nPm n(iξ) +Bm nQmn(iξ)]Pm n(η) [Emcosmθ+Fmsinmθ]. (1. 74) In the alternative form of Laplace’s equation the separated equations have solutions sin mθ,cosmθ,Pm n(cosβ),andPm n(isinhα),Qm n(isinhα).The par- tial solutions are similar to the form of (1. 74). 1.2.6 Elliptic cylinder coordinates The separated solutions of Laplace’s Equation (1. 38) in elliptic cylinder coordinates are ψ(ξ,η,z ) =A(α)B(β)Z(z) where, in general, AandBsatisfy Mathieu’s equation and the modiﬁed Math- ieu equation, respectively. For a full description of these functions and theirproperties, the reader is referred to [40] and [75]. If ψis independent of z, Laplace’s equation becomes ∂ 2ψ ∂α2+∂2ψ ∂β2= 0, which has separated solutions B(β) =B1 mcosmβ+B2 msinmβ, A(α) =A1 me−mα+A2memα. 1.2.7 Toroidal coordinates Our treatment of the method of separation of variables in this system is based on that given by N.N. Lebedev [36]. Unlike the cases considered previously, we cannot directly separate variables in Equation (1. 40). However, deﬁne a new function Vby ψ=V/radicalbig 2 coshα−2 cosβ, where√2 coshα−2 cosβmay be called the “asymmetry factor;” Laplace’s Equation (1. 40) becomes d2V dα2+d2V dβ2+ cothαdV dα+1 4V+1 sinh2αd2V dφ2= 0. ©200 1 CRC Press LLC This admits separation of variables: setting V=A(α)B(β)Φ(φ),we ﬁnd that sinh2α/bracketleftbigg1 Ad2A dα2+1 Bd2β dβ2+cothα AdA dα+1 4/bracketrightbigg =−1 Φd2Φ dφ2=µ2, whereµ2is a constant. This implies d2Φ dφ2+µ2Φ = 0, 1 Ad2A dα2+cothα AdA dα+1 4−µ2 sinh2α=−1 Bd2B dβ2=ν2, whereν2is another constant, so that d2B dβ2+ν2B= 0, 1 sinhαd dα/parenleftbigg sinhαdA dα/parenrightbigg −/parenleftbigg ν2−1 4+µ2 sinh2α/parenrightbigg A= 0. (1. 75) Thus Laplace’s equation in toroidal coordinates has inﬁnitely many partic- ular solutions of the form φ=/radicalbig 2 coshα−2 cosβAµ,ν(α)Bν(β)Φµ(φ), where Bν=Cνcos(νβ) +Dνsin(νβ), Φµ(φ) =Eµcos(µφ) +Fµsin(µφ), andA=Aµ,νsatisﬁes (1. 75). The introduction of a new variable z= coshα into this equation transforms it to d dz/bracketleftbigg (1−z2)dA dz/bracketrightbigg +/bracketleftbigg (ν−1 2)(ν+1 2)−µ2 1−z2/bracketrightbigg A= 0, which may be recognised as the diﬀerential equation for the associated Leg- endre functions Pµ ν−1 2orQµ ν−1 2; thus Aν,µ(α) =Gν,µPµ ν−1 2(coshα) +Hν,µQµ ν−1 2(coshα). 1.3 Formulation of potential theory for structures with edges The focus of this book is potential theory – the study of solutions of La- place’s equation – especially for structures in which edge eﬀects are important. ©200 1 CRC Press LLC Asalread yindicated ,theboundarycondition smus tbesupplementedbyade- cayconditionatinﬁnityaswellasﬁniteenergyconstraintsnearedges ,sothat auniqu eandp hysicallyrelevantsolutioncanbefound. Sinceedgesintr oducedistinctivefeature sintothetheor y,letusdistinguish betweenclose dsurfaces,thosepossessingnoboundar yoredge ,andopen shells,whichhaveoneormoreboundaries.Asphericalsurfacei sclosed ,whilst thehemisphericalshel lisope nwithacircularboundary.Amoresophisti- cateddistinctioncanbeformulatedi ntopologicalterms ,butthisisunneces- saryforourpurposes.Thesm oothnessofthesurface,includingthepresence ofsingularitiessuchascornersorconicaltips,isimportantinconsidering theexistenc eanduniquenessofsolutions.Thi stopichasbeenextensively investigatedbyKellogg[32].However ,thesurface sunde rinvestigationinthis bookareportionsofcoordinatesurfacesasdescribe dinth eIntr oduction,and boththesurface sandboundingcurvesareanalyti corpiecewis eanalytic .The smoothnessconditions,whichmustbeimposedontheclose doropensurfaces inamor egeneralformulationofpote ntialtheory,areautomaticall ysatisﬁed andwil lbeomittedfromfurtherdiscussionexceptfortwocases ,theconical shellsconsideredi nChapter6,andthetwo-dimensionalaxially-slotte dcylin- dersofarbitrarycross-sectionalproﬁleconsidere dinSection7.5;appropriate smoothnesscondition sareconsidere dintherespectivesections. Thissectionoutlinesgeneri caspectsofpote ntialtheor yapplicablet oboth open and closed surfaces, together with those features that are distinctive for open shells. Let us begin with the conditions under which a uniquenesstheorem, assuring existence of potentials for closed surfaces, can be asserted. A closed surface separates space into two regions, namely internal andex- ternal ; the internal region may be composed of two or more disconnected parts depending upon the topology of the closed surface. Thus, we can con- sider either the internal boundary value problem for Laplace’s equation or the external boundary value problem. The term boundary value problem requires an explicit deﬁnition of the type of boundary condition imposed on solutionsU(− →r) of Laplace’s equation on the closed surface S.EitherUis speciﬁed everywhere on S(the Dirichlet problem) or its normal derivative ∂U ∂n (in the direction of the outward normal− →nonS) is speciﬁed on S(the Neu- mann problem), or a linear combination of Uand its normal derivative is speciﬁed. These three types, known as ﬁrst-, second-, and third-kind bound-ary value problems, respectively, may be expressed as U=f 1onS, ∂U ∂n=f2onS, or∂U ∂n+h(U−f3) = 0 onS, ©200 1 CRC Press LLC wheref1, f2,f3,andhare given functions on S.Thus the internal Dirichlet boundary value problem for Laplace’s equation can be formulated as follows. Proble m1LetVbeagivenregionofspacewhichisopen,andisbound ed by the closed surface S.Find the function Uthat (a) satisﬁes Laplace’s equa- tion∆U= 0within the region V,(b) is continuous in the closed region V∪S including the boundary surface S,and (c) takes an assigned value on S. The external Dirichlet boundary value problem for an inﬁnite open region Vexterior to the closed surface Srequires an additional constraint on the behaviour of the solution as the observation point tends to inﬁnity.Proble m2 LetVbeaninﬁnit eopenregionexterio rtotheclosedsur- faceS. Find the function Uthat (a) satisﬁes Laplace’s equation ∆U= 0in the inﬁnite region V,(b) is continuous in the closed region V∪Sincluding the bounding surface S,(c) takes on assigned value on S,and (d) converges uniformly to zero at inﬁnity: U(− →r)→0as|− →r| → ∞. It is proved in [32] and [60] that when these conditions are satisﬁed, a unique solution providing a potential can be guaranteed. The Kelvin transform V(− →r) =r −1U(r−2− →r) ofUis harmonic, except at− →r=− →0 , ifUis harmonic (see [17])). If we require that the function Ube harmonic at inﬁnity, i.e., the function Vis harmonic at the origin, then condition (d) may be omitted; in either case, the radial derivative∂U/∂r =O(r−2) asr→ ∞ . Sometimes the conditions (a)–(c), or (a)–(d) above are referred to as “the conditions of the uniqueness theorem.”IfUis harmonic, and its value is prescribed on the surface S, thenVsolves the Dirichlet problem where its value is prescribed in the obvious way on the surfaceS /prime,which is the image of Sunder the Kelvin transform− →r/mapsto−→r−2− →r of inversion in a unit sphere centred at the origin. The strict demarcation of internal and exterior regions is lost once a closed surface is punctured and the potentials in previously disconnected regions are coupled to one another across the aperture introduced in the closed surface.Whilst the conditions described above are satisfactory for closed bodies, open surfaces require a supplementary condition to deal appropriately with the sin- gular behaviour of potentials near the edges or rims of the aperture boundarycurve. Physical motivation for the ﬁnal form and choice of this condition can be found in the electrostatic example of an ideally conducting body with a pointor edge. When charged, a high-level electrostatic ﬁeld is created near the point or edge due to charge concentration in its vicinity; the ﬁeld tends to inﬁnity as the point of observation approaches the point or edge. By contrast,away from the edge, the surface charge density varies smoothly as does the ©200 1 CRC Press LLC potential. However, in the vicinity of the edge, the electrostatic ﬁeld − →E=−∇U (1. 76) exhibits extremely high values. At ﬁrst sight, this localized high-level electrostatic ﬁeld might be considered an “equivalent source.” Nevertheless, some care is needed in this interpreta- tion because the energy integral attached to a real source occupying a volumeVdiverges: 1 2/integraldisplay/integraldisplay/integraldisplay Vε0/vextendsingle/vextendsingle/vextendsingle− →E/vextendsingle/vextendsingle/vextendsingle2 dV=∞. (1. 77) (As an illustration, consider a unit charge placed at the origin of a spherical coordinate frame. The potential is V=r−1and the electrostatic ﬁeld is radically directed:− →E=− →r/r3; the energy integral is clearly divergent.) On the other hand, the energy associated with the charged conductor might reasonably be expected to be ﬁnite, so that the apparent or equivalent sourcein the vicinity of the edge possesses a weaker (integrable) singularity thanthat of a real source. The discussion of appropriate models for real physical sources has a long history; suﬃce it to say that in the absence of such localized sources, the energy associated with the structure must remain bounded. This discussion provides a physical motivation for our additional “edge con- straint,” namely that the gradient of the potential (electrostatic or otherwise)must be square integrable over the whole volume Vof space: /integraldisplay/integraldisplay/integraldisplay V|gradU|2dV=/integraldisplay/integraldisplay/integraldisplay V|∇U|2dV <∞. (1. 78) Abstracting from the particular physical problem that the potential functionU(− →r) describes, we assume that the value |∇U| 2is proportional to the vol- ume density of the energy, and whereas this gradient may exhibit singularbehaviour at various points of the region under consideration, the total en-ergy within any bounded volume including the edges must be ﬁnite, as in (1.78). We will see later that this condition ensures that the potential is uniquelydetermined. From a mathematical point of view, the condition (1. 78) is important in establishing existence and uniqueness of solutions to Laplace’s equation. Oneway of demonstrating existence of solutions is via the “Dirichlet principle,”which asserts that any function Uthat minimises /integraldisplay/integraldisplay/integraldisplay V|gradU|2dV, (1. 79) subject to the constraint U=fonS,where the continuous function fis prescribed, satisﬁes Laplace’s equation ∆ U= 0 subject to the boundary conditionU=fonS.This principle has had a chequered career, which is traced in [43], but eventually it was placed on a rigorous basis for a large ©200 1 CRC Press LLC classofboundingsurfaces S.Theprinciplesti mulate dmuchcarefulanalysis ofsurfaces(ther earesurface sforwhichLaplace’ sequationcannotbesolved uniquely)andleadtoth edevelopmentoffunctionalanalysi sthroughth eex- aminationofth eclassoffunctionsforwhi chthemini mumof(1.79)i sactually attained. AcceptingthatLaplace’sequation,withth eboundarycondition U=f onS,hasatleastonesolution,uniquenessisestablishe dbyconsideringthe diﬀerenceU1ofanytwosuchdistinctsolutions. U1isharmonicandvanishes onS,andthedivergencetheoremshowsthat /integraldisplay/integraldisplay ΣU1∂U ∂rdS−/integraldisplay/integraldisplay SU1∂U ∂ndS=/integraldisplay/integraldisplay/integraldisplay V|gradU1|2dV, (1.80) whereΣdenote salargesphericalsurfaceofradiu sRenclosingS,and−→nis theoutwardnormalon S;thebounds ,U1=O(R−1)and∂U1 ∂r=O(R−2)as R→∞,showthatbot hsidesof(1.80)vanishas R→∞,sothatU1is identicallyzero,andthesolution Uisunique.Thisargume ntisno tdirectly validwhe nSisanopensurfac ewit hedge s(thedivergencetheoremisnot applicable);itmaybem odiﬁe dbysurroundin gtheopensurfacebyasmall openregionwithasmoothboundingsurface Sεwhosevolume εcontractsto zero;uniquenes shold sforthesurfac eSε,andbylettin gε→0,thesame resultcanberecoveredforthesurfac eS,providedtheenergyi ntegral(1.79) isﬁnite.Thesameidentitycanbeemployedtosh owthatifSisasmooth surfaceboundinganopenvolume,theenergyi ntegral(1.79)isﬁnite. ExamplesofnontrivialsolutionstoLaplace’sequationthatdecayatinﬁnity (accordingto U(−→r)→0as|−→r|→∞ )yetvanishonanopensurfac eS0may beconstructedshouldth erequirementofﬁnitenessoftheenergyi ntegralbe disregarded.Consider,incylindricalpolars( ρ,φ,z ),thehalf-plane φ=0.For anypositiveinteger n,thefunction sψn=Anρ−n 2sin(nφ/2)satisf yLaplace’s equation(witharbitraryconstants An)andvanis honS.Theimageof Sunder inversioninaunitspherelocatedat( ρ,φ,z )=(1,π,0)isacirculardis cD. TheKelvintransformof ψnisharmonicon D,vanisheson D,andisO(|−→r|−1) as|−→r|→∞. Thus ,infor mulatin gthestateme ntofboundar yvalueproblemsforLa- place’sequation,twodiﬀerencesbetweenclosedan dope nsurfacesar eappar- ent.First,thewell-deﬁnedconceptofinternalandexternalboundaryvalue problemsforclosedsurfacesdisappears ,thedeterminationofthepotentialfor opensurface sbecomesa mixed boundaryvalu eproblemforLaplace’sequa- tion;secondly,aswellasth econditionsstandardl yimposedinthedetermina- tionofth epote ntialﬁel dassociate dwithaclosedbody,anextraboundedness condition(1.78)mustbeimposedontheenergytodetermin euniquelythe potentialdistributionass ociate dwit hanope nsurface. Laterchaptersexamin epotentialtheor yforopenshellsthatareportions ofth eorthogonalc oordinatesurfacesdescribedi nSection1.1.B ywayof illustration, consider the particular example of a spherical shell S0of radiusa ©200 1 CRC Press LLC subtendinganangle θ0attheorigin ;itisdeﬁnedinsphericalc oordinatesby r=a,0≤θ≤θ0,0≤φ≤2π. Thesphericalsurfac eSofradiu samayberegardedastheunionofth eshell S0andthe“aperture” S1givenby r=a,θ 0<θ≤π,0≤φ≤2π. Proble m3 SupposetheshellSischargedtounitpotential .Findthe potentialU(r,θ,φ )thatsatisﬁesthefollowin gconditions:(1) ∆U=0atall points,exceptontheshell;(2) Uiseverywherecontinuous,includingallpoints onthesurface S=S0∪S1;onS0,Utakesaprescribedvalue: U(a,θ,φ )= Φ(θ,φ),atallpointsof S0;(3)thenormalorradialderivativeiscontinuous atallpointsof S1: lim r→a+∂U ∂r(r,θ,φ )=lim r→a−∂U ∂r(r,θ,φ )forθ0≤θ≤π,0≤φ≤2π; (4)Uconvergesuniformlyto 0atinﬁnity :U(r,θ,φ )→0asr→∞,and(5) theenergyintegralmustbebound edinanyvolum eVincludingtheedges: /integraldisplay/integraldisplay/integraldisplay V|∇U|2dV= /integraldisplay/integraldisplay/integraldisplay V/braceleftBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂U ∂r/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +1 r2/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂U ∂θ/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +1 r2sin2θ/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂U ∂φ/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/bracerightBigg dV<∞. Moregenerally,le tusformall ystatetheﬁrst-kin dmixe dboundaryvalue problem(BVP )forLaplace’ sequationpertainingtoanope nsurfaceS0that isaportionofacoordinat esurfac eSinoneofthos ecoordinatesystems inwhichLaplace’sequationca nbesolvedbythemeth odofseparationof variables(Section1.2).Theterm mixed referstotheenforceme ntofdiﬀerent boundary conditions on the two portions comprising the surface S(namely the shellS0and the aperture S1). Let (q1,q2,q3) be the curvilinear coordinates in this system, and suppose thatSis the coordinate surface on which q1takes a ﬁxed value, q0 1.LetI2and I3be the intervals over which q2andq3range (in the spherical cap example, I2= [0,π] andI3= [0,2π] whereq2andq3are identiﬁed with θandφ).Thus Sis parametrised by I=I2×I3. We consider shells S0which are parametrised by I0=I(0) 2×I(0) 3whereI(0) 2 is composed of one or more subintervals of I2,andI(0) 3is a similar subset of I3; however, as a rule, either I(0) 2=I2orI(0) 3=I3. The “aperture” area S1 may then be parametrised by I1,the complement of I0inI(I=I0∪I1). ©200 1 CRC Press LLC Proble m4 Theﬁrst-kin dmixedBVPforLaplace’sequation .Findthe potentialU=U(q1,q2,q3)satisfyingthefollowingconditions:(1) ∆U=0at allpoints,ofspaceexcepton S;(2)Uiseverywherecontinuous,includingall pointsonthesurface S=S0∪S1,thatis lim q1→q0 1+0U(q1,q2,q3)=lim q1→q0 1−0U(q1,q2,q3)(1.81) where (q2,q3)∈I;(3)thevalueof Uisprescribedon S0,byagivencontinuous functionF: lim q1→q0 1+0U(q1,q2,q3)=lim q1→q0 1−0U(q1,q2,q3)=F(q2,q3)(1.82) where (q2,q3)∈I0;(4)thenormalderivative∂U ∂q1mustbecontinuousonthe apertureS1: lim q1→q0 1+0∂U ∂q1(q1,q2,q3)=lim q1→q0 1−0∂U ∂q1(q1,q2,q3)(1.83) where (q2,q3)∈I1;(5)U(q1,q2,q3)convergesuniformlyto 0atinﬁnity: U(q1,q2,q3)→0as|(q1,q2,q3)|→∞ ;(1.84) and(6)theenergyint egralmustbeboundedi nanyarbitrar yvolumeVin- cludingtheedges: /integraldisplay/integraldisplay/integraldisplay V|∇U|2dV=/integraldisplay/integraldisplay/integraldisplay V/braceleftBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 h1∂U ∂q1/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 h2∂U ∂q2/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 h3∂U ∂q3/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/bracerightBigg dV<∞ (1.85) (whereh1,h2,h3arethemetriccoeﬃcients). Thecondition(1.85)gi vesrisetomos tofth esocalled edgeconditions appearin ginth eliterature ;theseprescribethesingularbehaviourofthepo- tentialclosetoanedge ,determining,forexample,theorde rofthesingularity. Itisworthnotingthatth enormalderi vativeiscontinuou sontothesurface S0,butmaytakediﬀerentvalue sasitsapproachesapointon S0fromone sideortheother .Also,i nthevicinityoftheedgeth enormalderivati veis generallyunbounded.Thejumpinnormalderivativeacros sthesurfac eS0 isthesingle-l ayerdensityusedinthestandar dintegralrepresentationofthe ﬁeld(seeSection1.7).P hysically,iti sproportionaltosurfacechargedensity. In contrast to ﬁrst-kind mixed problems are those of second kind, in which the role of Uand∂U ∂q1are interchanged in the boundary conditions (1. 82) and (1. 83). Proble m5Thesecond-kin dmixedBVPforLaplace’sequation .Findthe potentialU=U(q1,q2,q3)satisfying the following conditions: (1) ∆U= 0at ©200 1 CRC Press LLC all points of space, except on S; (2) the normal derivative ∂U ∂n=∂U ∂q1 is everywhere continuous, including all points on the surface S=S0∪S1,that is /bracketleftbigg∂U ∂q1/bracketrightbigg q1=q0 1−0=/bracketleftbigg∂U ∂q1/bracketrightbigg q1=q0 1+0(1. 86) where (q2,q3)∈I; (3) the value of the normal derivative is prescribed on S0, by a continuous function G: lim q1→q0 1−0∂U ∂q1(q1,q2,q3) = lim q1→q0 1+0∂U ∂q1(q1,q2,q3) =G(q2,q3), (1. 87) where (q2,q3)∈I0; (4)Uis continuous on the aperture S1: lim q1→q0 1−0U(q1,q2,q3) = lim q1→q0 1+0U(q1,q2,q3), (1. 88) where (q2,q3)∈I1; (5)U(q1,q2,q3)converges uniformly to 0as|(q1,q2,q3)| → ∞(cf. (1. 84)); and (6) the energy integral (1. 85) must be ﬁnite. Succeeding chapters provide constructive methods for uniquely solving both types of mixed boundary value problems for Laplace’s equation. Our methods utilise the special functions associated with the orthogonal coordinate systemof relevance to the particular problem at hand to obtain a pair of functionalequations, which are enforced on S 0and on the aperture S1, respectively. A constructive and rigorously correct mathematical method – to be explained inthe next chapter – may be applied to solve this pair, to determine completely the unique potential satisfying the appropriate six conditions listed above. Let us describe generally how these functional equations arise, for the ﬁrst- kind mixed boundary value problems for Laplace’s equation, under the some- what restrictive assumption that the solution is independent of one coordinate,sayq 3,so that ∂U ∂q3(q1,q2,q3) = 0. (1. 89) In this case the function F(see (1. 82)) is independent of q3:F(q2,q3)≡ F(q2). Dual (or multiple) series equations arise when the eigenvalue spectrum of the Sturm-Liouville problem, originating from the ordinary diﬀerential equa- tions obtained in application of the separation of variables technique applied to Laplace’s equation, is discrete. Separated solutions are generated for thetwo regions separated by S(namely, the regions q 1<qo 1andq1>qo 1) in the form Un(q1,q2) =/braceleftBigg x(1) nR(1) n(q1)An(q2), q1<qo 1 x(2)nR(2) n(q1)An(q2), q1>qo 1/bracerightBigg (1. 90) ©200 1 CRC Press LLC (where the index n= 0,1,2,...labels the spectrum), and the corresponding total solution is the superposition U(q1,q2) =∞/summationdisplay n=0Un(q1,q2) =∞/summationdisplay n=0/braceleftBigg x(1) nR(1) n(q1),q1<qo 1 x(2)nR(2) n(q1),q1>qo 1/bracerightBigg An(q2).(1. 91) The unknown Fourier coeﬃcients/braceleftBig x(1)n/bracerightBig∞ n=0and/braceleftBig x(2) n/bracerightBig∞ n=0are to be deter- mined;R(1) n,R(2) nareradial functions, and Anis an angle function by con- vention. BothR(1) nandR(2) nsatisfy the same ordinary diﬀerential equation and pro- vide a basis for the set of all solutions of this diﬀerential equation; R(1) nis chosen to be regular in the domain q1≤qo 1(so determining it uniquely up to a constant factor), whereas R(2) nis chosen to satisfy the condition (1. 84); thusR(2) nis regular in the domain q1≥qo 1and determined uniquely up to a constant factor. The inﬁnite set of angle functions {An}∞ n=0is complete and orthogonal on I2with respect to a weight function, denoted h: /integraldisplay I2h(q2)An(q2)Am(q2)dq2=αnδnm. (1. 92) The constants αnare necessarily positive, so that the normalised functions ˆAn=An/α1 2nform a complete orthonormal set. The continuity condition (1. 81), together with (1. 92), gives a relationship betweenx(1) nandx(2)n, x(2) n=/parenleftBig R(1) n(qo 1)/R(2) n(qo 1)/parenrightBig x(1)n, (1. 93) so that (1. 91) becomes U(q1,q2) =∞/summationdisplay n=0x(1)n/braceleftBigg R(1) n(q1), q 1<qo 1 R(1) n(qo 1)R(2) n(q1)/R(2) n(qo 1), q1>qo 1/bracerightBigg An(q2),(1. 94) or, in symmetric form, U(q1,q2) =∞/summationdisplay n=0Xn/braceleftBigg R(2) n(qo 1)R(1) n(q1), q1<qo 1 R(1) n(qo 1)R(2) n(q1), q1>qo 1/bracerightBigg An(q2), (1. 95) where we have rescaled x(1) n=R(2) n(qo 1)Xn.Enforcing the boundary conditions (1. 82) and (1. 83) leads to the pair of functional equations ∞/summationdisplay n=0XnR(1) n(qo 1)R(2) n(qo 1)An(q2) =F(q2), q2∈I(0) 2, (1. 96) ©200 1 CRC Press LLC ∞/summationdisplay n=0XnW/parenleftBig R(1) n(qo 1),R(2) n(qo 1)/parenrightBig An(q2) = 0, q2∈I2\I(0) 2, (1. 97) where the Wronskian W/parenleftBig R(1) n(q1),R(2) n(q1)/parenrightBig =R(1) n(q1)d dq1R(2) n(q1)−R(2) n(q1)d dq1R(1) n(q1) is evaluated at q1=qo 1.These equations are referred to as dual series equations if the interval I2\I(0) 2is a simply connected subset of I2; otherwise, they are referred to as triple - ormultiple -series equations depending on the total number of connected subintervals of I2appearing in Equations (1. 96) and (1. 97) Enforcement of the ﬁnite energy condition (1. 85) provides a unique solution to (1. 96) and (1. 97); essentially, it provides the correct functional space setting for the coeﬃcients Xn.The simplest but most eﬀective choice of the volumeVof integration in (1. 85) is the interior region ( q1≤qo 1,q2∈I2,q3∈ I3); it is bounded, ﬁnite, and involves the edges. Substitution of the relevant derivatives, obtained from term-by-term diﬀerentiation of (1. 91) and (1. 94),into the energy integral (1. 85) gives a condition which the Fourier coeﬃcients (x (1) norXn) must satisfy. This condition will always take the form ∞/summationdisplay n=0cn/vextendsingle/vextendsingle/vextendsinglex(1) n/vextendsingle/vextendsingle/vextendsingle2 <∞, (1. 98) wherecnis some explicitly known coeﬃcient. Conversely, as we will see in succeeding sections, the condition (1. 98) ensures that the operations of term-by-term integration and diﬀerentiation, to be applied on the series (1. 96) and (1. 97), are justiﬁed and valid. If theangle functions are normalised, the condition (1. 98) becomes ∞/summationdisplay n=0|yn|2<∞, (1. 99) where {yn}∞ n=0is a suitably rescaled sequence related to/braceleftBig x(1) n/bracerightBig∞ n=0or{Xn}∞ n=0. Thus the sequence {yn}∞n=0belongs to the set of square summable Fourier co- eﬃcientsl2. When the spectrum of the relevant Sturm-Liouville problem is continu- ous, a similar argument produces dual (or multiple) integral equations. This schematic outline of the formulation and basic features of boundary valueproblems for structures with edges will be reﬁned and analysed more care-fully when concrete conﬁgurations are encountered. ©200 1 CRC Press LLC 1.4 Dual equations: a classiﬁcation of solution methods It is perhaps well known that a comprehensive theory to solve dual equations does not exist, and that many treatments have been developed to obtain solutions to such equations. Essentially, these treatments can be groupedinto three basic methods: the deﬁnition-extension method described by W.E.Williams, [76] the substitution method described by B. Noble, [46] and themultiplying factor method also described by B. Noble [47]. The common and distinctive feature of all these methods is the utilization, in one form or another, of Abel’s integral equation (or transform) technique.Let us illustrate these methods with the simple problem of determining thepotential of a charged spherical cap when its surface is held at a constant unitvalue of potential. This problem produces the following dual series equationsinvolving Legendre polynomials P n(cosθ), ∞/summationdisplay n=0anPn(cosθ) = 1, θ∈(0,θ0), (1. 100) ∞/summationdisplay n=0(2n+ 1)anPn(cosθ) = 0, θ∈(θ0,π), (1. 101) where the unknown desired set of coeﬃcients {an}∞ n=0must belong to the Hilbert functional space l2. The concrete form of condition (1. 85) that imposes this constraint on the coeﬃcients is ∞/summationdisplay n=0n+ 1 2n+ 1|an|2<∞. (1. 102) 1.4.1 The deﬁnition method To solve Equations (1. 100) and (1. 101), let us deﬁne a function gon [0,θ0], which provides the extension of (1. 101) to the complete interval [0 ,π]. That is, let ∞/summationdisplay n=0(2n+ 1)anPn(cosθ) =/braceleftbigg g(θ), θ∈[0,θ0) 0, θ ∈(θ0,π]/bracerightbigg . (1. 103) In (1. 103) the left-hand side is the Fourier-Legendre expansion for a certain functionF; the right-hand side is the piecewise continuous expression of that functionFon [0,π]. The orthogonality property of the set of Legendre polynomials {Pn}∞ n=0on [0,π] allows us to express {an}∞n=0in terms of the function g: an=1 2/integraldisplayθ0 0g(θ)Pn(cosθ) sinθdθ. (1. 104) ©200 1 CRC Press LLC Substitute this expression in (1. 100), and invert the order of integration and summation; the original Equations (1. 100) and (1. 101) are then reduced tothe ﬁrst-kind Fredholm integral equation /integraldisplay θ0 0g∗(ϑ)K(ϑ,θ)dϑ= 1, θ∈(0,θ0) (1. 105) whereg∗(ϑ) = sinϑg(ϑ),and the kernel is K(ϑ,θ) =1 2∞/summationdisplay n=0Pn(cosϑ)Pn(cosθ). (1. 106) Following the idea developed by W. E. Williams [76], we solve (1. 105) by the successive solution of two Abel integral equations. To this end, represent the kernel (1. 106) in the form K(ϑ,θ) =1 2π/integraldisplaymin(ϑ,θ) 0dφ/radicalbig (cosφ−cosϑ)(cosφ−cosθ)(1. 107) This representation is easily obtained from the Dirichlet-Mehler formula (see AppendixB.94), Pn(cosθ) =√ 2 π/integraldisplayθ 0cos(n+1 2)φ√cosφ−cosθdφ, (1. 108) from which it follows that ∞/summationdisplay n=0Pn(cosϑ) cos(n+1 2)φ=/braceleftbigg [2(cosφ−cosϑ)]−1 2,0≤φ<ϑ 0, , ϑ<φ ≤π/bracerightbigg .(1. 109) Decomposing the integration domain in (1. 105) into two parts, (0 ,θ)∪(θ,θ0), and using the expression (1. 107), one obtains the repeated integral 1 2π/integraldisplayθ 0dφ√cosφ−cosθ/integraldisplayθ0 φg∗(ϑ)√cosφ−cosϑdϑ= 1, θ ∈(0,θ0). (1. 110) A double application of the inversion formulae to this iterated Abel integral equation (see the next section) yields the solution for the function gin closed form: g(ϑ) =√ 2 π/braceleftbigg2 cos1 2θ0√cosϑ−cosθ0+π 2−arcsin/parenleftbiggcos1 2θ0 cos1 2θ/parenrightbigg/bracerightbigg . (1. 111) The substitution of this expression for the function gin (1. 104) gives, after elementary integration, the ﬁnal solution for the Fourier coeﬃcients: an=1 π/bracketleftbiggsinnθ0 n+sin(n+ 1)θ0 n+ 1/bracketrightbigg . (1. 112) The function gcoincides with the surface charge density on the cap, and possesses the expected singularity of order −1 2asϑ→θ0(Formula (1. 111)). Moreover, by construction, the solution {an}∞ n=0given in (1. 112) lies in l2. ©200 1 CRC Press LLC 1.4.2 The substitution method This method is based upon expansion of a certain discontinuous function in a Fourier-Legendre series. Its analytic form permits us to ﬁnd a representation of the solution that automatically satisﬁes one of the dual equations. Con- sidering (1. 108), let us represent the coeﬃcients anin terms of an unknown functionU, so that an=/integraldisplayθ0 0U(t) cos(n+1 2)tdt. (1. 113) The functional Equation (1. 101) is automatically satisﬁed, but the companion Equation (1. 100) is transformed, after an interchange of integration andsummation, to the Abel integral equation /integraldisplay θ 0U(t)dt√ cost−cosθ=√ 2, θ ∈(0,θ0). (1. 114) This possesses the obvious solution U(t) =2 πcost 2, t ∈(0,θ0) and substitution in (1. 113) immediately leads to the previously obtained solution (1. 112). 1.4.3 Noble’s multiplying factor method The essence of the multiplying factor method is the following. Each of the Equations (1. 100) and (1. 101) is multiplied by a suitable functional factor and then an appropriate integral operator, or a combination of integral anddiﬀerential operators is applied to transform the left-hand side of (1. 100)or (1. 101) to the same functional expression – a Fourier series (or similar)involving the coeﬃcients a n.The coeﬃcients anare then obtained from the calculation of the Fourier coeﬃcients of the piecewise continuous function obtained by the transform of the right-hand side of (1. 100) and (1. 101) under this process. In our example problem, the operators are derived from the well-known identities arising from the inversion of the Dirichlet-Mehler formulae (B. 94): cos(n+1 2)θ=1√ 2d dθ/integraldisplayθ 0Pn(cosφ)√cosφ−cosθsinφdφ, (1. 115) cos(n+1 2)θ=1√ 2(n+1 2)/integraldisplayπ θPn(cosφ)√cosθ−cosφsinφdφ. (1. 116) LetK1andK2denote operators deﬁned by (K1f)(θ) =1√ 2d dθ/integraldisplayθ 0f(φ) sinφdφ√cosφ−cosθ(1. 117) ©200 1 CRC Press LLC and (K2f)(θ)=1√ 2/integraldisplayπ θf(φ)sinφdφ√cosθ−cosφ. (1.118) ApplyingK1to(1.100)and K2to(1.101)yields ∞/summationdisplay n=0ancos(n+1 2)θ=/braceleftbigg cos1 2θ,θ∈(0,θ0) 0,θ ∈(θ0,π)/bracerightbigg . (1.119) Acalculationofthecoeﬃcients an,usingtheorthogonalityproperty /integraldisplayπ 0cos(n+1 2)θcos(m+1 2)θdθ=π 2δnm, leadstoth epreviouslyobtaine dform(1.112)ofthedesiredsolution. Thusal lthethreemeth odsdescribedaboveemployAbel’sintegralequation insom eformoranother.Noble’smultiplyingfactormethodcanbeseen asadirectapplicatio noffractionalintegration.Th erelationshipbe tween fractionalintegrationandintegraltransformsofAbe ltypeisdiscussedfully in[55]. 1.4.4TheAbeli ntegraltransfor mmethod Amor edirectan dreadil yjustiﬁe dmethodofsolvin gdualserie sisthe Abel integra ltransformmethod whichwasdevelopedin[67],[68] ,[69],[70]and [71].Itcanbedirectl yide ntiﬁedwiththe integra lrep resentationmeth od describedin[21],theonlydiﬀerencebeingthatth emathematicalvalidity oftheoperation sintheﬁrstapproachisproperlyestablished ,whereasthe analysisofth elatte rapproachispurelyformalinmanner. Thisisnott oasser tthattheAbelintegraltransformmeth odisacompletely newmeth odtosolvedual ,triple,andmultipleseriesorintegralequationsof thisclass .Itisclearlyr ootedinthei ntegralreprese ntationmethoddescribed in[21].I tisworthemphasizingthateachofth esequence sofmathematical operationsass ociatedwiththeAbe lintegraltransformmeth odisstraightfor- wardl yjustiﬁed,sothereisnodoubtabou tthevalidityofsolutionsobtained bythi sapproach .Th enameofthemeth odhighlightsthetransfor matits core. Variousclassesofdualandtripleserie sequationsaresolvedinamathe- maticallyrigorousmanne rbytheapplicationofthismeth odinChapte r2. Bywayofillustration ,letusapplytheAbe lintegraltransformmeth odtothe chargedsphericalcapproble mdescribedearlierinthissection.Themethod transforms each functional equation (of the dual series equations) to an inte- gral equation that is recognizable as Abel’s integral equation with zero forcingterm. It has a unique solution, namely zero, which provides the basis for theﬁnal solution step. ©200 1 CRC Press LLC Considerin g(1.100),wereplac etheright-han dsidewiththeexpression, derivedfrom(1.108),n=0, 1=√ 2 π/integraldisplayθ 0cos1 2φdφ√cosφ−cosθ. Thetermsontheleft-han dsidearereplace dbythereprese ntation(1.108). Aninterchangeofintegratio nandsummatio nispermissibl eunderthecondi- tion(1.102),andleadstothehomogeneou sAbelintegra lequation /integraldisplayθ 0f(φ)dφ√cosφ−cosθ=0,θ∈[0,θ0), (1.120) where f(φ)=∞/summationdisplay n=0ancos(n+1 2)φ−cos1 2φ. Becaus eEquatio n(1.120)hastheuniqu ezerosolution ,weobtain ∞/summationdisplay n=0ancos(n+1 2)φ−cos1 2φ=0,φ∈[0,θ0). (1.121) Turnin gto(1.101),term- by-termintegratio nofthisseriesisalsopermit- ted,becaus etheseriesisuniforml yAbel-summabl e(thispointisdiscusse din greate rdetai linSectio n2.2).Multiplyin gbysinθandintegratin gover(θ,π) (whenθ>θ 0),produces ∞/summationdisplay n=0an[Pn−1(cosθ)−Pn+1(cosθ)] = 0, θ∈(θ0,π]. (1. 122) Herewehaveusedthewell-kn ownformula(seeAppendix ,(B.58)) (2n+ 1)Pn(x) =d dx[Pn−1(x)−Pn+1(x)], (1. 123) and the Dirichlet-Mehler formula (see [55]) for the Legendre polynomials Pn(cosθ) =√ 2 π/integraldisplayπ θsin(n+1 2)φdφ√cosθ−cosφ, (1. 124) to derive the Abel transform representation for the diﬀerence Pn−1(cosθ)−Pn+1(cosθ) =−2√ 2 π/integraldisplayπ θcos(n+1 2)φsinφdφ√cosθ−cosφ. (1. 125) Note that both Equations (1. 122) and (1. 100) possess a common feature. The asymptotics for the Legendre polynomials Pn(x) =O(n−1 2) asn→ ∞ ©200 1 CRC Press LLC ensure that the asymptotic behaviour of the terms in each series is the same. The substitution of the transform (1. 125) into (1. 122) produces the integral equation /integraldisplayπ θg(φ) sinφdφ√cosθ−cosφ= 0, θ∈(θ0,π], (1. 126) where g(φ) =∞/summationdisplay n=0ancos(n+1 2)φ, φ∈(θ0,π]; (1. 127) it has the solution ∞/summationdisplay n=0ancos(n+1 2)φ= 0, φ∈(θ0,π]. (1. 128) The interchange of summation and integration is justiﬁed under condition (1. 102). Combining the results (1. 121) and (1. 128) produces the same result as given by Noble’s multiplying factor method, and consequently the same closed form expression (1. 112) for the coeﬃcients an. In spite of its simplicity, this example illustrates all the features that are characteristic of the Abel integral transform method. The main features thatoccur in a typical application to potential theory, which requires the solutionof dual, triple, or multiple series equations, or integral equations of this type,the kernels of which involve hypergeometric functions, are as follows. The very ﬁrst step is to determine the solution class from the edge condition. This key point allows us to establish the validity of various mathematical operations on series or integrals. Next, the convergence rate of each member of the dual equations must be assessed. For example, the convergence rateof (1. 100) is O(n −3 2) asn→ ∞,whereas the rate of (1. 101 ) is O(n−1 2), asn→ ∞.The equation with the slower convergence is subjected to an integration operation that equilibrates the convergence rate of both equations (for example, see the transition from (1. 101) to (1. 122)). Although both members of this pair of transformed functional equations now possess the same convergence rate, each involves diﬀerent kernels ( Pn(cosθ) orPn−1(cosθ)−Pn+1(cosθ) in the example above). The third step represents the kernel (and right-hand side) of each equation as an Abel integral trans-form. One can then interchange the order of summation and integration (fordual series equations), or the order of double integrals (for dual integral equa- tions), as appropriate. As a result, one obtains two independent integral equations of Abel type, each of which possesses a unique solution, namelyzero (see (1. 121) and (1. 128) of the example above). The ﬁnal phase is to recognize that a Fourier series (or Fourier integral) in the unknown coeﬃcients has been obtained by this process; the series or ©200 1 CRC Press LLC integralisequaltoaknownfunction,withapiecewis econti nuou sreprese nta- tiononitscomplet eintervalofdeﬁnition .Ourexampleabovepr oducedthe representation(1.119).Usingpropertiesoforthogonali tyandcompleteness ofterm sintheseries–orani nvers eFourie rintegraltransformasappropriate –weobtaintheﬁnalsolutionfortheunkn ownc oeﬃcie nts. Thisgeneraldescriptionprovide sacommonformalstructur etoconstruct solutionsofmultipleseriesorintegralequations.Th emathematicaltoolsto realiseeachstepofthi sprocessarediscussedinChapter2. 1.5 Abel’s integral equation and Abel integral trans- forms Most texts on linear integral equations invariably discuss Abel’s integral equation in the ﬁrst few pages because it is a precursor of the modern theory of linear integral equations. Originally, Abel’s integral equation f(x) =/integraldisplayx 0u(ξ)dξ√x−ξ(1. 129) arose from the following problem in mechanics. A particle moving underthe inﬂuence of gravity, along a smooth curve in a vertical plane, takes thetimef(x) to move from the vertical height xto a ﬁxed point on the curve. The problem is to ﬁnd the function udeﬁning that curve, known as the tautochrone. Instead of Equation (1 .129),Abel set himself the problem of solving the more general equation f(x) =/integraldisplay x au(ξ)dξ (x−ξ)λ, (0<λ< 1), (1. 130) wherefis a known function and uis the function to be determined. Details of the solution of this generalised Abel’s equation can be found in several texts, including [63], [50], and [24]; we simply state the inversion formula for (1.130): u(ξ) =sinλπ πd dξ/integraldisplayξ af(x)dx (ξ−x)1−λ. (1. 131) The companion form of the generalised Abel integral equation is f(x) =/integraldisplayb xu(ξ)dξ (ξ−x)λ, (0<λ< 1), (1. 132) and has the solution u(ξ) =−sinλπ πd dξ/integraldisplayb ξf(x)dx (x−ξ)1−λ. (1. 133) ©200 1 CRC Press LLC Inthedeductio nof(1.131)and(1.133),thefollowingwellknownformula involvin gthebetafunctio nB(seeAppendix ,(B.8))isused, /integraldisplayz ξdx (z−x)1−µ(x−ξ)µ=B(µ,1−µ) =π sinµπ, (0<µ< 1). (1. 134) From the solution given in [24] we may formulate a theorem concerning solution existence. Theorem 1 Necessary and suﬃcient conditions that the integral Equation (1.130) should have a continuous solution on (a,b)are thatf(x)be continuous in(a,b), thatf(a) = 0 , and that /integraldisplayx af(ξ)dξ (x−ξ)1−λ have a continuous derivative on (a,b). If these conditions are fulﬁlled, (1.130) has only one continuous solution, given by Formula (1.131). An analogous theorem may be stated for the integral Equation (1 .132). Omitting their deduction (see [55]), let us state three results connected with Abel’s integral equation, which will be used subsequently. Theorem 2 Ifφis ﬁnite, and has only a ﬁnite number of discontinuities in (a,b), the function Φ(x) =/integraldisplayx aφ(ξ)dξ (x−ξ)λ, (λ<1), is continuous on (a,b), including at the point a, where it vanishes. Theorem 3 Ifφis continuous on (a,b),and has a derivative that is ﬁnite except for a ﬁnite number of discontinuities in (a,b), and ifφ(a) = 0 , the function Φ(x) =/integraldisplayx aφ(ξ) (x−ξ)λdξ, (λ<1), has a derivative that is continuous on (a,b)and is given by the formula Φ/prime(x) =/integraldisplayx aφ/prime(ξ) (x−ξ)λdξ. Theorem 4 (Dirichlet’s extended formula). Let φbe a function of two vari- ables. Ifφis ﬁnite in the region a≤y≤x≤b,and its discontinuities (if any) are regularly distributed, and if λ,µ,ν are constants satisfying 0≤λ<1,0≤µ<1,0≤ν <1, ©200 1 CRC Press LLC then /integraldisplayb a/integraldisplayx aφ(x,y)dydx (x−y)λ(b−x)µ(y−a)ν=/integraldisplayb a/integraldisplayb yφ(x,y)dxdy (x−y)λ(b−x)µ(y−a)ν. (1. 135) Let us consider a further generalisation of Abel’s integral equation in the form f(x) =/integraldisplayx aU(ξ)dξ {h(x)−h(ξ)}λ, x ∈(a,b), 0<λ< 1, (1. 136) wherehis a strictly monotonically increasing and continuously diﬀerentiable function on ( a,b) (soh/prime>0 in this interval). Diﬀering terminology has been used for this generalisation in the literature. R. P. Kanwal [31] treated the generalised Abel integral Equation (1 .130) or (1.132) as a special case of the singular integral equation (1.136). I. N. Sneddon [55] refers to (1 .136) as an Abel-type integral equation, but usually in the context of some speciﬁc choices of the function h. We propose to use this terminology whatever choice for h is made. The pairs (1 .130) and (1 .131), or (1.132) and (1 .133), can be considered as companion integral transforms. For example, if the transform (1 .130) is designated as the direct Abel integral transform, then integral transform(1.131) is its inverse. Similar terminology can be applied to the pair (1 .132) and (1.133). Let us solve Equation (1 .136),following the treatments [50] and [55] closely. Consider the integral/integraldisplay x ah/prime(u)f(u)du {h(x)−h(u)}1−λ, and substitute for ffrom (1.136) to obtain /integraldisplayx a/integraldisplayu aU(ξ)h/prime(u)dξdu {h(u)−h(ξ)}λ{h(x)−h(u)}1−λ. By changing the order of integration, this becomes /integraldisplayx aU(ξ)dξ/integraldisplayx ξh/prime(u)du {h(u)−h(ξ)}λ{h(x)−h(u)}1−λ. The inner integral reduces to (1 .134) under the obvious change of variable z=h(u), so that /integraldisplayx ah/prime(u)f(u)du {h(x)−h(u)}1−λ=π sinλπ/integraldisplayx aU(ξ)dξ. (1. 137) Diﬀerentiation of both sides of (1 .137) produces the solution U(ξ) =sinλπ πd dξ/integraldisplayξ ah/prime(u)f(u)du {h(ξ)−h(u)}1−λ. (1. 138) ©200 1 CRC Press LLC Similarly, the integral equation f(x) =/integraldisplayb xU(ξ)dξ {h(x)−h(u)}λ, x ∈(a,b), 0<λ< 1 (1. 139) has the solution U(ξ) =−sinλπ πd dξ/integraldisplayb ξh/prime(u)f(u)du {h(u)−h(ξ)}1−λ. (1. 140) Two special cases of (1 .136) and (1 .139) will be of further interest. First, leth(ξ) =ξ2: the integral equation f(x) =/integraldisplayx aU(ξ)dξ (x2−ξ2)λ, (0<λ< 1) (1. 141) has the solution U(ξ) =2 sinλπ πd dξ/integraldisplayξ auf(u)du (x2−ξ2)1−λ, (1. 142) while its companion f(x) =/integraldisplayb xU(ξ)dξ (ξ2−x2)λ, (0<λ< 1) (1. 143) has the solution U(ξ) =−2 sinλπ πd dξ/integraldisplayb ξuf(u)du (µ2−ξ2)1−λ. (1. 144) Next, consider h(ξ) = coshξ: the integral equation f(x) =/integraldisplayx aU(ξ)dξ (coshx−coshξ)λ, (0<λ< 1) (1. 145) has the solution U(ξ) =sinλπ πd dξ/integraldisplayξ asinhuf(u)du (coshξ−coshu)1−λ, (1. 146) while the companion integral equation f(x) =/integraldisplayb xU(ξ)dξ (coshξ−coshu)λ, (0<λ< 1) (1. 147) has the solution U(ξ) =−sinλπ πd dξ/integraldisplayb ξsinhuf(u)du (coshu−coshξ)1−λ. (1. 148) ©200 1 CRC Press LLC 1.6Abel-typeintegra lrepres entation sofhypergeomet- ricfunctions InSectio n1.3weencou nteredthereprese ntationofonetypeofhypergeo- metri cfunctio nasanintegra ltransfor mofAbeltype.TheDirichlet-Mehler formulaeprovideanintegra lreprese ntationfortheLegendr epolynomial s(see (1.124)andAppendix ,(B.94)),expresse dintrigonometri cformas Pn(cosθ) =√ 2 π/integraldisplayθ 0cos(n+1 2)φ√cosφ−cosθdφ, (1. 149) Pn(cosθ) =√ 2 π/integraldisplayπ θsin(n+1 2)φ√cosθ−cosφdφ. (1. 150) At ﬁrst glance it seems that representations (1. 149) and (1. 150) transform one class of functions (the Legendre polynomials Pn(cosθ)) to another, the trigonometric functions of form cos( n+1 2)θ,and sin(n+1 2)θ. From a wider perspective, these functions may be regarded as members of one and the same class, namely the Jacobi polynomials P(α,β) n. For each ﬁxed (α,β),withα>1,β >−1,the Jacobi polynomials P(α,β) n are polynomials of degreen(= 0,1,2...) and are orthogonal on [ −1,1] with respect to the weight functionwα,β(x) = (1 −x)α(1 +x)β.Their properties are discussed in Ap- pendix B.3. In particular, the relations between the trigonometric functions and the Legendre polynomials are cosnθ=Γ/parenleftbig1 2/parenrightbig Γ (n+ 1) Γ/parenleftbig n+1 2/parenrightbigP(−1 2,−1 2) n (cosθ), (1. 151) cos(n+1 2)θ=Γ/parenleftbig1 2/parenrightbig Γ (n+ 1) Γ/parenleftbig n+1 2/parenrightbigcos1 2θP(−1 2,1 2) n (cosθ), (1. 152) sinnθ=Γ/parenleftbig3 2/parenrightbig Γ (n+ 1) Γ/parenleftbig n+1 2/parenrightbigsinθP(1 2,1 2) n−1(cosθ), (1. 153) sin(n+1 2)θ=Γ/parenleftbig1 2/parenrightbig Γ (n+ 1) Γ/parenleftbig n+1 2/parenrightbigsin1 2θP(1 2,−1 2) n (cosθ), (1. 154) and Pn(cosθ) =P(0,0) n(cosθ). (1. 155) On the other hand, the trigonometric functions cos( νx),sin(νx) with con- tinuous parameter ν, occur in the well-known representations [19] of the Bessel functions J0(νρ) =2 π/integraldisplayρ 0cosνx/radicalbig ρ2−x2dx, (1. 156) ©200 1 CRC Press LLC J0(νρ)=2 π/integraldisplay∞ ρsinνx/radicalbig x2−ρ2dx. (1.157) Echoingpreviou sremark saboutthereprese ntation s(1.149)and(1.150), anAbel-typetransfor mofthetrigonometri cfunction s(cosνx,sinνx)pro- ducesanothe rfunctiona lclass(J0).However,uponrecallin gthewell-kn own relationship s[19] cosνx=/parenleftBigπνx 2/parenrightBig1 2J−1 2(νx), (1.158) sinνx=/parenleftBigπνx 2/parenrightBig1 2J1 2(νx), (1.159) itbecome sclearthattheAbeltransform s(1.156)and(1.157)shoul dbe considere dinthewidercontextofBesse lfunctions. Inotherwords,thetrigonometri cfunction scosnθ,sinnθ,cos(n+1 2)θ,and sin(n+1 2)θ,withintegerorhalf-i ntegerparameter ,shoul dbeconsidere das aspecialsubclassoftheJacob ipolynomial sP(α,β) n(cosθ)(forappropriate (α,β));wherea sthetrigonometri cfunction scosνx,sinνx,withrealparame- terν,shoul dbeconsidere dasaspecialsubclassoftheBesse lfunction sJµ(νx) (forappropriat eµ). Inturn,boththeclassofBesse lfunction sJµandtheclassofJacob ipoly- nomial sP(α,β) n,witharbitrar yvaluesoftheparameter s(α,β)orµ,belongto thewiderclassofhypergeometri cfunction sinaverysimpl emanner .Both areparticula rexample softhegeneralise dhypergeometri cfunctio n[59] pFq(a1,...,ap;b1,...,bq;z)≡∞/summationdisplay k=0(a1)k(a2)k....(ap)k (b1)k(b2)k...(bq)k·zk k!(1.160) wherethenotatio nforthePochhamme rsymbol (a)kdef=a(a+1)...(a+k−1);(a)0def=1 (1.161) hasbeenused;theupperparameters−→a=(a1,...,ap)areunrestricted, wherea sthelowerparameters−→b=(b1,...,bq)arerestricte dsothatnobjis zerooranegati veinteger .Notethatwhenaisneithe rzeronoranegati ve integer, (a)k=Γ(a+k) Γ(a). (1.162) Whenp≤q,theseriesconvergesforallcomple xz;whenp=q+1,the serieshasradiu sofconvergenc e1(itsconvergenc eontheunitdisc|z|=1 isdiscusse dinAppendixB.2).Iftheoneofupperparameter sisequalto zero or a negative integer, then the series terminates and is a hypergeometric polynomial. ©200 1 CRC Press LLC TheJacob ipolynomia lP(α,β) n mayberecognise dasageneralise dhyperge- ometri cfunctio n(seeAppendix ,(B.25));itishypergeometri cpolynomial P(α,β) n(x)=/parenleftbiggn+α n/parenrightbigg 2F1/parenleftbigg −n,n+α+β+1;α+1;1−x 2/parenrightbigg .(1.163) Fromthesymmetr yproperty(seeAppendix ,(B.26)) P(α,β) n(−x)=(−1)nP(β,α) n(x), wededuc ethealternati vereprese ntation P(α,β) n(x)=(−1)n/parenleftbiggn+β n/parenrightbigg 2F1/parenleftbigg −n,n+α+β+1;β+1;1+x 2/parenrightbigg . (1.164) Besse lfunction sofarbitrar yorderalsohaveahypergeometri creprese nta- tionintermsofthespecialconﬂuen thypergeometri cfunctions, Jµ(z)=(z/2)µ Γ(µ+1)0F1(µ+1;−1 4z2), (1.165) Jµ(z)=(z/2)µ Γ(µ+1)eiz 1F1(µ+1 2;2µ+1;2iv). (1.166) Letusderivetheintegra lreprese ntationofAbeltypefortheJacob ipoly- nomials .From(1.163)and(1.160)immediatel yfollowstheﬁniteseries represe ntation: P(α,β) n(x)=Γ(n+α+1) n!Γ(α+1)n/summationdisplay m=0(−n)m(n+α+β+1)m m!(α+1)m/parenleftbigg1−x 2/parenrightbiggm . (1.167) Fixtheparamete rη∈[0,1);multipl ybothsidesof(1.167)bythefactor (1−x)α(x−t)−ηandintegrat eovertheinterval(t,1)toobtain /integraldisplay1 t(1−x)αP(α,β) n(x) (x−t)ηdx =Γ(n+α+1) n!Γ(α+1)n/summationdisplay m=0(−n)m(n+α+β+1)m 2mm!(α+1)mAm+α η(t),(1.168) where Aq η(t)def=/integraldisplay1 t(1−x)q(x−t)−ηdx. (1.169) Thechangeofvariabl eby1−x=(1−t)yexpresse sAq η(t)intermsofthe betafunctio nB(seeAppendix ,(B.8)): Aq η(t) = (1 −t)q+1−η/integraldisplay1 0yq(1−y)−ηdy= (1−t)q+1−ηB(q+ 1,1−η) = (1−t)q+1−ηΓ(q+ 1)Γ(1 −η) Γ(q+ 2−η). (1. 170) ©200 1 CRC Press LLC Substituting (1. 170) into (1. 168), replacing ( α,β) by (α+η−1,β−η+ 1), and bearing in mind Deﬁnition (1. 167), one obtains, after some manipulation, the following integral representation of Abel type : P(α,β) n(t) =(1−t)−αΓ(n+ 1 +α) Γ(1−η)Γ(n+α+η)/integraldisplay1 t(1−x)α+η−1P(α+η−1,β−η+1) n (x) (x−t)ηdx. (1. 171) Interchanging the role of αandβin (1. 171), changing the sign of xand t,and taking into account Identity (1. 164), we obtain another such integral representation: P(α,β) n(t) =(1 +t)−βΓ(n+ 1 +β) Γ(1−η)Γ(n+β+η)/integraldisplayt −1(1 +x)β+η−1P(α−η+1,β+η−1) n (x) (t−x)ηdx. (1. 172) Formulae (1. 171) and (1. 172) have an interpretation in terms of fractional integration operators [55]. When η= 0,the following two notable identities corresponding to integration in conventional sense result: (1−t)α+1P(α+1,β−1) n (t) = (n+α+ 1)/integraldisplay1 t(1−x)αP(α,β) n(x)dx, (1. 173) (1 +t)β+1P(α−1,β+1) n (t) = (n+β+ 1)/integraldisplayt −1(1 +x)βP(α,β) n(x)dx. (1. 174) When expressed in algebraic form, the Dirichlet-Mehler Formulae (1. 149) and (1. 150) are special cases of the integral representations (1. 171) and (1. 172) withα=β= 0,η=1 2(settingt= cosθ,andx= cosφ): Pn(t) =π−1 2Γ(n+ 1) Γ(n+1 2)/integraldisplay1 t(1−x)−1 2P(−1 2,1 2) n (x) (x−t)1 2dx, (1. 175) Pn(t) =π−1 2Γ(n+ 1) Γ(n+1 2)/integraldisplayt −1(1 +x)−1 2P(1 2,−1 2) n (x) (t−x)1 2dx. (1. 176) Let us now obtain the integral representations of Abel kind for the Bessel functions. The well-known Sonine’s integrals provide a simple starting point. Sonine’s ﬁrst integral [14] is Jν+ξ+1(z) =zξ+1 2ξΓ(ξ+ 1)/integraldisplayπ 2 0Jν(zsinθ) sinν+1θcos2ξ+1θdθ, (1. 177) whereν >−1,ξ>−1.The trivial transformation z=xt,ρ =xsinθproduces the desired integral representation of Abel kind : t−ξ−1Jν+ξ+1(xt) =x−ξ−ν−1 2ξΓ(ξ+ 1)/integraldisplayx 0Jν(ρt)ρν+1(x2−ρ2)ξdρ. (1. 178) ©200 1 CRC Press LLC Alimitin gformofSonine’ssecon dintegral[55]is t−η−1Jν−η−1(xt)=xν−η−1 2ξΓ(η+1)/integraldisplay∞ 0Jν[t(s2+x2)1 2](s2+x2)−ν 2s2η+1ds, (1.179) whereν 2−1 4>η> −1.Thesubstitution s2+x2=ρ2transforms(1.179)to thesecond integra lrepresentationofAbelkin dforBesse lfunctions: t−η−1Jν−η−1(xt)=xν−η−1 2ηΓ(η+1)/integraldisplay∞ xJν(ρt)ρ−ν+1(ρ2−x2)ηdρ. (1.180) Specialcase sof(1.178)an d(1.180)with ξ=0an dη=0,respecti velyare 1 tJν+1(xt)=x−ν−1/integraldisplayx 0Jν(ρt)ρν+1dρ, (1.181) 1 tJν−1(xt)=xν−1/integraldisplay∞ xJν(ρt)ρ−ν+1dρ. (1.182) Thecommentsaboutfractionalintegrationdirectlyfollowin gFormul a(1. 172)areofequalpertinenc etothereprese ntations(1.178)an d(1.180)and theirconﬂuentforms(1.181)and(1.182). ThesebasicintegralrepresentationsofAbe lkindwillbeextensivel yex- ploitedinlate rchapters.Othe rusefu lrelationshipscanbefoun din[55]. 1.7Dualequationsandsingle -ordouble-layersurface potentials LetS0beanopensurface ,whichisaportiono falargerclose dsurfac eS; letS1beth ecompleme ntarypartof S0inS(thusS=S0∪S1)sothatS1may beregarde dasan“aperture”i nS.GivenS0,thechoic eofS(andhence S1) maybemadearbitrarily,butweshal lrequirethati tsatisﬁesthehy potheses fortheapplicationofGreen’stheorem(se e[32]). Classicalpote ntialtheoryreprese ntsthesolutionofLaplace’sequationby meansofsingle-ordouble-layersurfac epote ntials[32].InSection1.3,thefor- mulationofmixedboundar yvalueproblem sforS0andtheLaplaceequation wasdiscussed.Thi sapparentlyalternativeapproach(whichproducesdual seriesequation sorduali ntegralequations)isinfacte ntirelyequivalent,at leasti nthecontextofth eclassofcoordinatesurface sSdiscussedi nSection 1.3. LetPbe an arbitrary point on S, andMbe an observation point. Introduce an originO; let− →r/primeand− →rdenote the position vectors− − →OPand− − →OM, and denote the distance between PandMbyR PM=R/parenleftBig− →r/prime,− →r/parenrightBig =/vextendsingle/vextendsingle/vextendsingle− →r−− →r/prime/vextendsingle/vextendsingle/vextendsingle.AtP, we ©200 1 CRC Press LLC shallalsoconside rtheinward -andou tward-pointin guni tnormalvectors−→ni and−→ne. Letuscommenc ebyconsideringth eﬁrstboundaryvalueproblemforLa- place’sequation ,assumin gthatth evalu eofpotential Uisspeciﬁe donthe opensurfaceS0:U(−→r/prime)=F(−→r/prime)forsomecontinuousfunction F.Asalready mentioned,classicalpotentialtheor ypresentsthesolutionofLaplace’sequa- tionintermsofsurfac epote ntials.AsaconsequenceofGreen’sfundamental theorem[32],th evalueofth eharmonicfunction Uatanyinteriorpoint−→rof theregionboundedby Sisgivenby U(i)(−→r)=1 4π/integraldisplay/integraldisplay S/bracketleftBigg 1 R(−→r,−→r/prime)∂U(i) ∂ni−U(i)(−→ r/prime)∂ ∂ni/parenleftBigg 1 R(−→r,−→r/prime)/parenrightBigg/bracketrightBigg ds. (1.183) When−→rliesoutsid eS,theintegralin(1.183)vanishes.Intheexterior region,thesolutionatanypoint−→rexteriorto Ssatisﬁes U(e)(−→r)=1 4π/integraldisplay/integraldisplay S/bracketleftBigg 1 R(−→r,−→r/prime)∂U(e) ∂ne−U(e)(−→ r/prime)∂ ∂ne/parenleftBigg 1 R(−→r,−→r/prime)/parenrightBigg/bracketrightBigg ds. (1.184) When−→rliesinsideS,theintegralin(1.184)vanishes. Whenthesurfaceisopen ,thedistinctionbe tweeninterna landexternal regionsdisap pears(seeSection1.3)andthesolutionatanypoint−→rnoton Smust be considered as a sum of (1. 183) and (1. 184), U(− →r) =U(i)(− →r) +U(e)(− →r). (1. 185) The solution and its normal derivative must be continuous at any point− →r/primeof the aperture surface S1so that U(i)(− → r/prime)−U(e)(− → r/prime) = 0, (1. 186) ∂ ∂nU(i)(− → r/prime)−∂ ∂nU(e)(− → r/prime) = 0, (1. 187) where− →n≡− →ne=−− →ni. Thus the solution Uof the ﬁrst-kind boundary value problem is given by U(− →r) =−1 4π/integraldisplay/integraldisplay S0/bracketleftbigg∂U(i) ∂n−∂U(e) ∂n/bracketrightbigg1 R(− →r,− →r/prime)ds, (1. 188) whereas the solution of the second-kind boundary value problem (in which the normal derivative is speciﬁed on S0) is represented by U(− →r) =−1 4π/integraldisplay/integraldisplay So/bracketleftBig U(e)(− → r/prime)−U(i)(− → r/prime)/bracketrightBig∂ ∂n/parenleftBigg 1 R(− →r,− →r/prime)/parenrightBigg ds. (1. 189) ©200 1 CRC Press LLC Introducing the notations for the jump functions occurring in (1. 188 ) and (1. 189), σD(− → r/prime)def=∂U(i) ∂n−∂U(e) ∂n, (1. 190) σN(− → r/prime)def=U(e)(− → r/prime)−U(i)(− → r/prime), (1. 191) the integral formulae become U(− →r) =−1 4π/integraldisplay/integraldisplay S0σD(− → r/prime)1 R(− →r,− →r/prime)ds, (1. 192) and U(− →r) =−1 4π/integraldisplay/integraldisplay S0σN(− → r/prime)∂ ∂n/parenleftBigg 1 R(− →r,− →r/prime)/parenrightBigg ds. (1. 193) The ﬁrst integral (1. 192) is the potential associated with a simple or single- layer distribution on S; the second integral (1. 193) is the potential of a double-layer distribution on S[32]. Thus the ﬁrst-kind boundary value problem, in which the Dirichlet bound- ary condition (prescribing the value of UonS0) is given by U|S0=F, gives rise to the following Fredholm integral equation of the ﬁrst kind for the un-known single-layer distribution σ D: F(− → r/prime s) =−1 4π/integraldisplay/integraldisplay S0σD(− → r/prime)1 R(− →rs,− →r/prime)ds,− →rs∈S0. (1. 194) In a similar way, the second-kind boundary value problem in which the Neu- mann boundary condition (prescribing the value of∂U ∂nonS0) is given by ∂U ∂n|S0=Gproduces a Fredholm integral equation of the ﬁrst kind for the unknown double-layer distribution σN: G(− →rs) =−1 4π/integraldisplay/integraldisplay S0σN(− → r/prime)∂2 ∂ns∂n/prime/bracketleftBigg 1 R(− →rs,− →r/prime)/bracketrightBigg ds− →rs∈S0(1. 195) where− →nsdenotes the outward-pointing unit normal at− →rs. The distance function, between any two arbitrary points in space− →rand− →r/prime, R(− →r,− → r/prime)≡/vextendsingle/vextendsingle/vextendsingle− →r−− → r/prime/vextendsingle/vextendsingle/vextendsingle plays an important part in classical potential theory since the Green’s function for Laplace’s equation in three-dimensional free space is G(− →r,− → r /prime) =1 4π1 R(− →r,− →r/prime). (1. 196) ©200 1 CRC Press LLC Thereciprocalofthedistanc efunction ,R−1(−→r,−→r/prime),isofte ncalle dthesource functionofLaplace’sequation ;itisth epotentialfunctionassociatedwiththe positiveunitchargeinelectrostatics.Itsol vesthenon-homogeneou sLaplace’s equation(Poisson’sequation) ∇2U(−→r)=−δ(−→r−−→ r/prime), (1.197) whereδ(−→r−−→r/prime)isthedelta-function[28] ;thediﬀerentiationin(1.197)is performedwithrespec ttoth eprime dvariables. Wewishtoinvestigatepotentialproblemsincoordinat esystem sthatadmit separationofvariablesforLaplace’sequation.Accordingly,letusconsider Poisson’sequationingeneralise dcurvilinearc oordinates( q1,q2,q3): ∇2U(q1,q2,q3)=−h−1 q1h−1 q2h−1 q3δ(q1−q/prime 1)δ(q2−q/prime 2)δ(q3−q/prime 3)(1.198) wherehqi(i=1,2,3)ar ethemetriccoeﬃcients(seeSection1.1)and,as before, the diﬀerentiation in (1. 198) is performed with respect to the primedvariables. The metric coeﬃcients perform a normalising function in (1. 198)because /integraldisplay Vδ(− →r−− → r/prime)dV=/integraldisplay/integraldisplay/integraldisplay allq1,q2,q3δ(− →r−− → r/prime)hq1hq2hq3dq1dq2dq3= 1,(1. 199) which follows from the fundamental property of the δ-function, /integraldisplayx/prime=x+ε x/prime=x−εδ(x/prime−x)dx= 1,forε>0. We wish to obtain the Fourier series, or Fourier integral representation as appropriate, for the source function or for the Green’s function. We considerin detail the spherical coordinate context, and simply state the ﬁnal resultsfor other coordinate systems. In spherical coordinates ( r,θ,φ ) the Green’s functionG 0(− →r,− →r/prime) of free space must satisfy ∆G0(r,θ,φ,r/prime,θ/prime,φ/prime) =−1 r2sinθδ(r−r/prime)δ(θ−θ/prime)δ(φ−φ/prime), (1. 200) where the Laplacian operator ∆ is given by (1. 27). Since G0(− →r,− →r/prime) sat- isﬁes the homogeneous Laplace’s equation when− →r/negationslash=− →r/prime, and is a sym- metric function of the primed and unprimed coordinates, we may expand G0(r,θ,φ ;r/prime,θ/prime,φ/prime) in terms of eigenfunctions of the Laplacian as 1 r/prime∞/summationdisplay m=0cosm(φ−φ/prime)∞/summationdisplay n=mAnmPm n(cosθ)Pm n(cosθ/prime)/braceleftbigg(r/r/prime)n, r<r/prime (r/r/prime)−n−1, r>r/prime/bracerightbigg . (1. 201) This function is ﬁnite at r= 0 and satisﬁes the regularity condition at inﬁnity. ©200 1 CRC Press LLC Thevalu eAnmisdeterminedbytheinhomogeneoustermof(1.200). Multiplybot hside softhisequationby r2,andi ntegratewithrespec ttor overasmallinterval( r/prime−/epsilon1,r/prime+/epsilon1)abou tr/prime.Rememberingth econtinuityof thetermsat r=r/primeandpassingtothelimi t/epsilon1→0,weobtain r2∂ ∂rGo(r,θ,φ ;r/prime,θ/prime,φ/prime)|r=r/prime+0 r=r/prime−0=−1 sinθδ(θ−θ/prime)δ(φ−φ/prime). (1.202) Substituting(1.201)inthisexpression,andutilisin gtheWronskianrelation fortheinde pendentsolutionsof(1 .59),weﬁnd ∞/summationdisplay m=0cosm(φ−φ/prime)∞/summationdisplay n=mAnm(2n+1)Pm n(cosθ)Pm n(cosθ/prime) =δ(θ−θ/prime)δ(φ−φ/prime) sinθ.(1.203) Multiplyingbothside softhisequatio nbyPk l(cosθ)coskφandintegrating overthefullrangeofthevariables θandφproduces Anm=1 4π(2−δm0)(n−m)! (n+m)!. (1.204) Therepresentation(1.201)ofth efreespac eGreen’sfunctionwithcoeﬃ- cients(1.204)i snotuniqueinsphericalcoordinates.Itisarepresentation thatis discontinuou sinth ecoo rdinater.Arepresentationthatisdiscontin- uousinthecoordinate θwillbederivedi nChapte r6. Similar representations of the free-space Green’s function may be deduced by this method for those coordinate systems where the method of separation of variables is applicable. In particular, let us now state the Green’s functions ofthis type for the Laplace equation in Cartesian, cylindrical polar, and sphericalcoordinates. Cartesia ncoordinates. The distance function is R(− →r,− → r /prime) =/braceleftBig (x−x/prime)2+ (y−y/prime)2+ (z−z/prime)2/bracerightBig1 2, (1. 205) and the Green’s function Go(x,y,z ;x/prime,y/prime,z/prime),which is discontinuous in z,is 2 π2/integraldisplay∞ 0dνcos[ν(x−x/prime)]/integraldisplay∞ 0cos[µ(y−y/prime)]/radicalbig ν2+µ2/braceleftBigg e−√ ν2+µ2(z−z/prime), z>z/prime e√ ν2+µ2(z−z/prime), z<z/prime/bracerightBigg . (1. 206) ©200 1 CRC Press LLC Cylindrica lpolarcoordinates. The distance function is R(− →r,− → r/prime) =/braceleftBig ρ2+ (ρ/prime)2−2ρρ/primecos(φ−φ/prime) + (z−z/prime)2/bracerightBig1 2, (1. 207) and the Green’s function Go(ρ,φ,z ;ρ/prime,φ/prime,z/prime),which is discontinuous in ρ,is 1 π2/integraldisplay∞ 0dνcos[ν(z−z/prime)]× ∞/summationdisplay m=0(2−δ0m) cosm(φ−φ/prime)/braceleftbiggIm(νρ)Km(νρ/prime), ρ<ρ/prime Im(νρ/prime)Km(νρ), ρ>ρ/prime/bracerightbigg .(1. 208) Spherica lpolarcoordinates. The distance function is R(− →r,− → r/prime) =/braceleftBig r2+ (r/prime)2−2rr/prime[cosθcosθ/prime+ sinθsinθ/primecos(φ−φ/prime)]/bracerightBig1 2, (1. 209) and the Green’s function Go(r,θ,φ ;r/prime,θ/prime,φ/prime),which is discontinuous in r,is given by Formulae (1. 201) and (1. 204). Let us now establish the equivalence of the “dual series approach” and the method of single- or double-layer potentials in solving mixed boundary value problems for Laplace’s equation. A constructive proof is not very complicated, requiring the three steps outlined below. First, the free-space Green’s function for Laplace’s equation is expanded as a Fourier series, or represented as a Fourier integral, as in (1. 201), (1. 206), or (1. 208). Secondly, the unknown distributions σD(− →r/prime) orσN(− →r/prime) are also expanded in a Fourier series or as a Fourier integral. On the surface, S=S0∪S1,the jump functions introduced in (1. 190) and (1. 191) satisfy ∂U(i) ∂n−∂U(e) ∂n=/braceleftBigg σD(− →r/prime),onS0 0, onS1/bracerightBigg (1. 210) and U(e)(− → r/prime)−U(i)(− → r/prime) =/braceleftBigg σN(− →r/prime),onS0 0, onS1/bracerightBigg . (1. 211) These expansions are substituted in the integral Equations (1. 194) and (1. 195); because of the relationships of (1. 210) and (1. 211), the surface of integration is extended to the whole of S, which we may suppose is the coor- dinate surface corresponding to one coordinate (say q1) being held constant, whilst the remaining two coordinates q2,q3are varied over their full interval of deﬁnition. On the surface S,the harmonic functions (which are the sep- arated solutions of Laplace’s equation) are orthogonal and form a complete ©200 1 CRC Press LLC basis. Multiplying both sides of these equations by such a surface harmonic, and integrating over S(i.e., over the complete interval of variation of q2,q3), we obtain functional equations in matrix or integral form; these are valid forthose values of q 2,q3such that (q1,q2,q3)∈So.Additional equations are de- rived from Formulae (1. 210) and (1. 211) deﬁning the jump functions on theaperture surface S 1. Let us illustrate this abstractly described process with a concrete example. Consider the ﬁrst-kind boundary value problem for Laplace’s equation posedon an open spherical surface S 0(or spherical cap) of radius a, subtending an angleθ0at the origin, with the boundary condition on S0being given as U|S0=F. LetSandS1denote, respectively, the complete spherical surface of radiusa,and the aperture r=a, θ<θ 0≤π,0≤φ≤2π. OnS,the potential function U(a,θ,φ ) given by the function F(θ,φ) is expressible as a Fourier series F(θ,φ) =∞/summationdisplay m=0(2−δ0m) cosmφ∞/summationdisplay n=mam nPm n(cosθ), (1. 212) where am n=1 2π/integraldisplay2π 0dφ/integraldisplayπ 0dθsinθ.F(θ,φ)Pm n(cosθ) cosmφ (1. 213) are known Fourier coeﬃcients. We shall ﬁnd the solution of the Laplace equation in this case as single-layer potential (1. 192). Expand the jump function (1. 210) in spherical surface harmonics /bracketleftbigg∂U(i) ∂r−∂U(e) ∂r/bracketrightbigg r=a=1 a∞/summationdisplay m=0(2−δ0m) cosmφ/prime∞/summationdisplay n=mxm nPm n(cosθ/prime),(1. 214) whereθ/prime∈[0,π],φ/prime∈[0,2π],and{xm n}∞,∞ m=0,n=mdenotes its unknown Fourier coeﬃcients. We substitute the Green’s function, G0(a,θ,φ ;a,θ/prime,φ/prime) given by (1. 201) with r=r/prime=a, into (1. 194) to ﬁnd F(θ,φ) =−a2/integraldisplay2π 0dφ/prime/integraldisplay∞ 0dθ/primesinθ/primeσD(θ/prime,φ/prime)Go(a,θ,φ ;a,θ/prime,φ/prime).(1. 215) This is valid for θ∈[0,θ0), φ∈[0,2π]. Using (1. 210) and expansions (1. 212) and (1. 214), we obtain from (1. ©200 1 CRC Press LLC 215) the Fredholm integral equation of ﬁrst kind 1 4π/integraldisplay2π 0dφ/prime/integraldisplayπ 0dθ/primesinθ/prime/braceleftBigg∞/summationdisplay m=0(2−δm0) cosmφ/prime∞/summationdisplay n=mxm nPm n(cosθ/prime)/bracerightBigg ×∞/summationdisplay s=0(2−δs0) coss(φ−φ/prime)∞/summationdisplay l=s(l−s)! (l+s)!Ps l(cosθ/prime)Ps l(cosθ) =−∞/summationdisplay m=0(2−δm0) cosmφ∞/summationdisplay n=mam nPm n(cosθ).(1. 216) This is valid for θ∈[0,θ0), φ∈[0,2π]. Exploiting the orthogonality of spherical surface harmonics, the left-hand side of this equation simpliﬁes to a double series of the same format as the right-hand side, leading ﬁnally to the series equations ∞/summationdisplay m=0(2−δ0m) cosmφ∞/summationdisplay n=mxm n 2n+ 1Pm n(cosθ) =−∞/summationdisplay m=0(2−δ0m) cosmφ∞/summationdisplay n=mam nPm n(cosθ).(1. 217) This is also valid for θ∈[0,θ0), φ∈[0,2π]. A companion equation follows directly from the deﬁnition of jump function (1. 210) and its expansion in spherical surface harmonics: ∞/summationdisplay m=0(2−δ0m) cosmφ∞/summationdisplay n=mxmnPm n(cosθ) = 0. (1. 218) This is valid for the range θ∈(θ0,π], φ∈[0,2π]. Multiplication of both sides of Equations (1. 217) and (1. 218) by the factor coskφ, followed by integration with respect to φon [0,2π],produces a pair of dual series equations for the unknown coeﬃcients xm n: ∞/summationdisplay n=mxmn 2n+ 1Pm n(cosθ) =−∞/summationdisplay n=mam nPm n(cosθ),θ∈[0,θ0),(1. 219) ∞/summationdisplay n=mxm nPm n(cosθ) = 0, θ ∈(θo,π].(1. 220) Conversely, it is evident that transformation of the dual series equations (1. 217) and (1. 218) to an integral equation of Fredholm type can be easily realised in the following way. Apply the formula (1. 214) in reverse order,i.e., forφ /prime∈[0,2π], ∞/summationdisplay m=0(2−δm0) cosmφ/prime∞/summationdisplay n=mxm nPm n(cosθ/prime) =/braceleftbiggaσD(θ/prime,φ/prime), θ/prime∈[0,θ0) 0, θ/prime∈(θ0,π],(1. 221) ©200 1 CRC Press LLC from which it immediately follows that xm n=a(2n+ 1) 4π(n−m)! (n+m)!× /integraldisplay2π 0dφ/primecosmφ/prime/integraldisplayθ0 0σD(θ/prime,φ/prime)Pm n(cosθ/prime) sinθ/primedθ/prime.(1. 222) Substitution of (1. 222) in (1. 217) and an interchange of the order of sum- mation and integration produces the original integral Equation (1. 215), asdesired. Thus we have demonstrated the equivalence of the integral equation formu- lation ((1. 194) or (1. 215)) and the dual series equations formulation ((1.219) and (1. 220)) for determining the potential. ©200 1 CRC Press LLC Chapter2 SeriesandIntegralEquations Thespatialdistributionoftheelectrostaticpotentialsurroundin gaconduct- ingsurface(ope norclosed)isdetermine datthemostfundamentalle velby Laplace’sequation ,togethe rwiththeappropriateboundar yconditions,dec ay conditionsatinﬁnity,and,i fnecessar y,edgeconditions.Theprecis eformu- lationoftheseconditionswasdescribedinSection1.3. Analternativebutequi valentformulationutilize sintegralreprese ntations forth epotentialinterm softhesurfac echargedensity(corres pondin gtothe jumpinth enormalderi vativeofth epote ntialacrossth esurface);i nturn, thisdensityi sdeterminedasthesolutionofanintegralequationholdingat eachpointofth econductingsurface(seeSection1.7). Thesetwoformulation sareth ebasi sofal lanalyticalandnumericalmeth- odsdevise dtosolvethepote ntialproblemforbodiesofarbitrar yorgeneral shape .Certainclassesofsurfaces,includingthosethatareportionsofth e orthogonalcoordinatesurfacesdescri bedinChapter1,admi tanother formu- lation of the potential problem, in terms of dual- (or triple- or multiple-) series equations, or dual- (or triple- or multiple-) integral equations. Although it isformally equivalent, this alternative approach has the beneﬁt that, in many cases of physical interest, these equations can be solved analytically (in closed form), so that a direct assessment of the eﬀect of edges and cavities in thesegeometries is possible. In other cases, the analytical solution process trans-forms or regularises the series (or integral) equations to a matrix (or integral)Fredholm equation of the second kind. Once converted, these equations pro- vide a basis for approximate analytical solution techniques (such as successive approximation), or for a numerical solution procedure which is simple to im-plement, well conditioned, rapidly converging, and of guaranteed accuracy.Thus, edge eﬀects and cavity contributions to the potential distribution canbe accurately quantiﬁed. Beyond the electrostatic context, this approach ﬁnds general application to mixed boundary value problems (of ﬁrst-, second-, or third-kind) for theLaplace equation. It also provides a basis for assessing the scattering anddiﬀraction by the class of bodies described above, of acoustic and electromag-netic waves, where the interest is in accurate quantiﬁcation of the scatteringprocess by edges, or of entrapment of wave energy by cavities. This chapter considers various classes of series and integral equations. The core idea is to convert the set of equations to a second-kind Fredholm matrix ©200 1 CRC Press LLC orintegralequation.TheAbelintegraltransformmeth odprovidesauniﬁed andconstructivetreatme ntofthisprocess.Insom ecasestheseequation scan besolvedexplicitl y,inclosedform;inth eremainingcases,thetransformed systemi swellsuitedtoeithe rapproximateanalyticalsolutionmethodsorto numericalmethods.Whenthesecond-kin dmatrixsyste mistruncate dtoa ﬁnitenu mberNtroflinearequations,th esolutio nofth etruncate dsystem converge stotheexactsolutionas Ntr→∞ .Itispossibletoestimateaccuracy asafunctionoftruncationnumber Ntrandsoproducesolutionsofs peciﬁed accuracy.Precisetreatme ntsofth ebeh aviou rofsecond-kindsystem sunder truncationaregivenin[2]and[30]. Proof softhevalidi tyofthismethod ,andoftheuniquenessofsolutions, aresketchedinSection2.1;reader swithadeeperinterestinthedetail sare recommendedtoconsultth epaper[64]. Theproblemtypiﬁe dbythedeterminationoftheelectrostaticpotential surroundingacharge dsphericalcap(Section1.3)leadst odualseriesequa- tionsi nvolvingth eJacob ipolynomial sP(α,β) n askernels .Thi sgeneralclass ofequationsistheﬁrsttobeconsideredi nthenextsection.The yhavethe form∞/summationdisplay n=0cnxnP(α,β) n(t)=F(t),t∈(−1,t0), (2.1) ∞/summationdisplay n=0xnP(α,β) n(t)=G(t),t∈(t0,1), (2.2) wherethefunctions F,G andc oeﬃcientscnareknown, t0isﬁxedin( −1,1), andtheunknownc oeﬃcie ntsxnaretobedetermined .Typically, cn=n2η/parenleftbig 1+O(n−1)/parenrightbig ,asn→∞. TheregularisationgenerallyobtainedbytheAbeltransformmethodisout- lined,andwher epossible ,explici tsolution sarefound. Twos pecialsubclasse swhichmeritsomeseparateconsiderationar eexam- inedinthefollowin gtwosections(2. 2and2.3),dualserieswithtrigonometric kernelsorwithass ociatedLegendr efunctionkernel s(thesearecloselyrelated toultrasphericalpolynomials). Tripleseriesequationsprovideanaturalgeneralisationofdualseriesequa- tions;th ekernelclassexaminedi nSection2. 4isrestrictedtothos ekernel sof interesti nsubsequentchapters. Preparatorytoconsideringdualintegralequationsinthei rownright,the relationshipbe tweenseriesandintegralequation sisexploredinSection2.5. ThefollowingSection(2.6)demonstrateshowtoapplyth eAbe lintegraltrans- formtosol vesomedualintegralequationswithBesselfunctionkernels;this allowsu storegulariseawideclassofsuchduali ntegralequations. Thesubdivisionofth eintervalofdeﬁnitionfortripleserie sequationsex- aminedinSection2.4isassume dtobesymmetric ;thisrestrictionisrem oved tocoverasymmetricsu bdivision sinSection2.7. ©200 1 CRC Press LLC Couple dsystem sofseriesequation saretreate dintheSectio n2.8,and some general remarks on so-called integro-series equations are provided in the concluding section of the chapter. 2.1 Dual series equations involving Jacobi polynomials This section considers dual series equations of the form (2. 1) and (2. 2). Since the function y=P(α,β) n satisﬁes the diﬀerential equation 1 w(x)d dx/parenleftbigg/parenleftbig 1−x2/parenrightbig w(x)dy dx/parenrightbigg +n(n+α+β+ 1)y= 0, with weight function w(x) = (1 −x)α(1 +x)β, the parameter ηmay be as- sumed to lie in the interval [0 ,1); for ifη≥1,we may replace P(α,β) n(t) by −1 n(n+α+β+ 1)1 w(t)d dt/parenleftBigg /parenleftbig 1−t2/parenrightbig w(t)dP(α,β) n dt(t)/parenrightBigg and integrate twice to obtain an equation similar to (2. 1), but with a new coeﬃcientcnsatisfying cn=n2(η−1)/parenleftbig 1 +O(n−1)/parenrightbig ,asn→ ∞. It is convenient to employ the quantity λn(α,β;η) given by λn(α,β;η) =Γ (n+α+ 1) Γ (n+β+ 1 +η) Γ (n+α+ 1−η) Γ (n+β+ 1)(2. 3) whereΓdenote stheGamma-function ;Field’ sformula(seeAppendix ,(B.7)) shows that λn(α,β;η) =n2η/bracketleftbig 1 +O/parenleftbig n−1/parenrightbig/bracketrightbig . We consider the slightly more general form of (2. 1) and (2. 2): ∞/summationdisplay n=0λn(α,β;η)xn(1−rn)P(α,β) n(t) =F(t), t∈(−1,t0) (2. 4) ∞/summationdisplay n=0xn(1−qn)P(α,β) n(t) =G(t), t∈(t0,1). (2. 5) The inﬁnite set of unknown coeﬃcients {xn}∞ n=0are to be determined. The parameters α,β,η are constrained to satisfy α−η >−1,β >−1,and for our applications we may always suppose that η∈(0,1).The reason for this ©200 1 CRC Press LLC constrai ntwillbecom eclearoncethemeth odofregularisatio nisdescri bed below.Thequantities{rn}∞ n=0,{qn}∞n=0areassume dtobeknownsequences, ingeneral ,ofcomple xquantitiessatisfying lim n→∞qn=lim n→∞rn=0. (2.6) Theright-han dsidesofEquation s(2.4)and(2.5)areassume dtobeexpand- ableinFourier-Jacob iseriesoftheform F(t)=∞/summationdisplay n=0λn(α,β;η)fnP(α,β) n(t), (2.7) G(t)=∞/summationdisplay n=0gnP(α,β) n(t). (2.8) Weseeksolution sto(2.4)and(2.5)inanappropriat efunctiona lspace. Denot ebyl2(µ)thespaceofsequence s{xn}∞ n=0satisfying ∞/summationdisplay n=0nµ|xn|2<∞. (2.9) Wesupposethatthecoeﬃcie ntsfn,gnbelongtol2(2η−1),andthesolution willbesoughtinthesameclass: {xn}∞ n=0∈l2(2η−1),{fn,gn}∞n=0∈l2(2η−1). (2.10) Thespeciﬁcatio n(2.9)arisesverynaturall yinconnectio nwiththeedgecon- ditionoftheuniquenes stheore mforanopensurfac e(seeSectio n1.3,(1.85)). Thus(2.5)and(2.8)containseriesthatconvergetotheirsumsinthe weightedmeansquar esensewithweightw. Somecareisneede dintheinterpretatio nofconvergenc eoftheseriesoccur- ringin(2.4)and(2.7).Inourapplications ,Equatio n(2.4)invariabl yarises fromenforcin gthecontinuityofeitherthepotentialorofitsnorma lderivative acros stheapertur esurfac eofthestructur eunderconsideration .Thus(2.4) issummabl einthesenseofAbel(seeAppendixD.2),andmore overbecause thecoeﬃcie ntsin(2.4)areO(nr)forsomerdependentonlyonα,β,and uniforml yon[−1,t0],theseriesisuniforml yAbel-summabl eon[−1,t0]. Letusnowdescri bethegenera lschemetosolveEquation s(2.4)and(2.5)by theprocessofregularisation .Thiswasbrieﬂ youtline dattheendofSection 1.4.Firstweintegrate ,withweightfunctio n(1+t)β,bothsidesof(2.4)over the interval ( −1,t), using the integration formula (1 .174) to obtain a more rapidly converging series: ∞/summationdisplay n=0λn(α,β;η) (n+ 1 +β){xn(1−rn)−fn}P(α−1,β+1) n (t) = 0,t∈(−1,t0) (2. 11) ©200 1 CRC Press LLC This process is justiﬁed because the series is uniformly Abel-summable (on closed subintervals of ( −1,t0)). Next we use the integral representation (1 .172) of Abel kind for Jacobi polynomials P(α−1,β+1) n,replacing index αbyα−1 andβbyβ+ 1: P(α−1,β+1) n (t) = (1 +t)−β−1Γ (n+β+ 2) Γ (1−η) Γ (n+β+ 1 +η)/integraldisplayt −1(1 +x)β+ηP(α−η,β+η) n (x) (t−x)ηdx(2. 12) Substituting this representation for P(α−1,β+1) n in (2.11) and interchanging the order of summation and integration, we obtain the functional equation /integraldisplayt −1(t−x)−ηU(x)dx= 0,t∈(−1,t0) (2. 13) where U(x) = (1 +x)β+η∞/summationdisplay n=0Γ (n+α+ 1) Γ (n+α+ 1−η)[xn(1−rn)−fn]P(α−η,β+η) n (x). (2. 14) In obtaining the last equation, deﬁnition (2 .3) was used. The interchange is justiﬁed by the weighted mean square convergence of the series (2 .14) (see AppendixD.2).Thereaso nfortheconstrai ntα−η>−1,β>−1isnow clear. Equation (2 .13) is the homogenous form of Abel’s integral equation. The inverse formula (1 .131) shows that (2 .13) has the unique trivial solution, and we obtain the functional equation ∞/summationdisplay n=0Γ (n+α+ 1) Γ (n+α+ 1−η)[xn(1−rn)−fn]P(α−η,β+η) n (x) = 0,x∈(−1,t0). (2. 15) To obtain a second equation over the interval ( t0,1),involving the same Jacobi polynomials as in (2. 15), it is necessary to utilise the integral repre- sentation (1. 171), replacing ηby 1−η: P(α,β) n(t) =(1−t)−αΓ (n+ 1 +α) Γ (η) Γ (n+α+ 1−η)/integraldisplay1 t(1−x)α−ηP(α−η,β+η) n (x) (x−t)1−ηdx. (2. 16) Repeating the mathematical operations used to obtain equation (2 .15) we ﬁnd ∞/summationdisplay n=0Γ (n+α+ 1) Γ (n+α+ 1−η){xn(1−qn)−gn}P(α−η,β+η) n (x) = 0,x∈(t0,1). (2. 17) ©200 1 CRC Press LLC Combining Equations (2 .15) with (2 .17) we obtain ∞/summationdisplay n=0cnxnP(α−η,β+η) n (x) =/braceleftbigg F1(x), x∈(−1,t0) F2(x), x∈(t0,1)/bracerightbigg (2. 18) where F1(x) =∞/summationdisplay n=0cn(xnrn+fn)P(α−η,β+η) n (x), F2(x) =∞/summationdisplay n=0cn(xnqn+gn)P(α−η,β+η) n (x), and cn=Γ (n+α+ 1) Γ (n+α+ 1−η). We recall that the coeﬃcients {xn}∞ n=0lie in a space dependent upon η (2.10). It simpliﬁes the solution to modify the Fourier coeﬃcients so that they are square-summable sequences in l2=l2(0).Introducing the orthonormal Jacobi polynomials ˆP(α,β) n,deﬁned by ˆP(α,β) n(z) =P(α,β) n(z)//bardblP(α,β) n/bardbl (2. 19) wherethesquar enorm/bardblP(α,β) n/bardbl2≡h(α,β) n is given by Formula (B. 20) (see Appendix ),wemaynormalis ethecoeﬃcie ntsxn,fn,gnso that {yn,ˆfn,ˆgn}=Γ (n+ 1 +α) Γ (n+ 1 +α−η)/bracketleftBig h(α−η,β+η) n/bracketrightBig1 2{xn,fn,gn}; (2. 20) these sequences are square-summable: {yn,ˆfn,ˆgn}∞ n=0∈l2(0)≡l2. (2. 21) Equation (2 .18) becomes ∞/summationdisplay n=0ynˆP(α−η,β+η) n (t) =/braceleftbigg G1(t), t∈(−1,t0) G2(t), t∈(t0,1)/bracerightbigg , (2. 22) where G1(t) =∞/summationdisplay n=0(ynrn+ˆfn)ˆP(α−η,β+η) n (t), G2(t) =∞/summationdisplay n=0(ynqn+ ˆgn)ˆP(α−η,β+η) n (t). ©200 1 CRC Press LLC Conditions (2 .7), (2.8),and (2.10) dictate that all series in (2 .22) are Fourier-Jacobi series, so that we can exploit completeness and orthogonal- ity of the orthonormal set {ˆP(α−η,β+η) n }∞ n=0on [−1,1]. After multiplication of both sides of (2 .22) by the factor (1 −t)α−η(1 +t)β+ηˆP(α−η,β+η) s (t) and inte- gration over ( −1,1),we obtain the following inﬁnite system of linear algebraic equations (i.s.l.a.e.) (1−rs)ys+∞/summationdisplay n=0yn(rn−qn)ˆQ(α−η,β+η) sn (t0) =ˆfs+∞/summationdisplay n=0/parenleftBig ˆgn−ˆfn/parenrightBig ˆQ(α−η,β+η) sn (t0),(2. 23) wheres= 0,1,2,..., and ˆQ(α,β) sn(t) =/integraldisplay1 t(1−x)α(1 +x)βˆP(α,β) s (x)ˆP(α,β) n (x)dx. (2. 24) The function ˆQ(α,β) sn(t) is termed an incomplete scalar product of normalised Jacobi polynomials with weight function (1 −x)α(1 +x)βfor the following reason. The conventional (weighted) scalar product of ˆP(α,β) s and ˆP(α,β) n is ˆQ(α,β) sn(−1) =/integraldisplay1 −1(1−x)α(1 +x)βˆP(α,β) s (x)ˆP(α,β) n (x)dx (2. 25) and the “incompleteness” of (2 .24) refers to the fact that integration is per- formed over the subinterval [ t,1]. We shall also employ the unnormalised incomplete scalar product Q(α,β) sn(t) =/integraldisplay1 t(1−x)α(1 +x)βP(α,β) s (x)P(α,β) n (x)dx (2. 26) of unnormalised Jacobi polynomials. Some useful properties incomplete scalar productarestatedinAppendixB.6. It can be shown that {ˆQ(α,β) sn(t)}∞ s,n=0is the matrix of a projection operator K(t) inl2, therefore satisfying K(t)2=K(t).Using this property and that of the diagonal operators dranddqwhich correspond to the diagonal matrices diag{rn}∞ n=0and diag {qn}∞n=0, one can prove that the matrix operator of (2.23) is a completely continuous (or compact) perturbation Hof the identity operatorIinl2. Thus Equation (2 .23) is a Fredholm equation of the second kind(seeAppendixC.3),whichwemayreprese ntintheform (I−H)y=b (2. 27) where the vector b∈l2may be readily identiﬁed; the solution vector y= {yn}∞n=0lies inl2.Since projection operators have norm at most unity, the ©200 1 CRC Press LLC norm of the operator His bounded by /bardblH/bardbl≤max n|rn|+ max n|qn|. (2. 28) TheFredhol malternati ve(seeAppendixC.3)isvalidfor(2.23)or(2.27); the equations can be solved by the truncation method or, in certain cases, by an iterative method of successive approximations. The truncation method re- places the inﬁnite system by a ﬁnite number (those indexed by s= 0,1,...,N tr) of linear algebraic equations, in which all inﬁnite sums are truncated to retain only the variables y0,y1,...,y Ntr.Note that the solution is explicitly obtained in closed analytic form when rn=qn= 0.The mixed boundary value prob- lems considered later will either have analytic solutions of this type or havesolutions which can, in principle, be obtained by the method of successive approximations. In any case the system (2 .23) is solvable numerically in a satisfactory manner via the truncation method. A detailed discussion of the rate of convergence of the solution to the truncated system to the exact (inﬁ-nite) system is given in [30]; this makes it possible to estimate and guaranteeaccuracy of numerical solutions generated in this fashion. A companion pair to the Equations (2 .4) and (2.5) is the related set of dual series ∞/summationdisplay n=0xn(1−qn)P(α,β) n (t) =G(t),t∈(−1,t0) (2. 29) ∞/summationdisplay n=0λn(β,α,η )xn(1−rn)P(α,β) n (t) =F(t),t∈(t0,1). (2. 30) The indices α,β,η are now constrained to satisfy α>−1,β−η>−1.Essen- tially, the subintervals on which (2 .4) and (2.5) are enforced are interchanged, and the factor λn(β,α,η ) replacesλn(α,β,η ).In contrast to (2 .7),Fis as- sumed to be expandable in a Fourier-Jacobi series of the form F(t) =∞/summationdisplay n=0λn(β,α,η )fnP(α,β) n (t), (2. 31) butGis assumed to possess the same expansion (2. 8). Applying the same method used above to solve (2 .4) and (2.5),we ﬁnd ∞/summationdisplay n=0cnxnP(α+η,β−η) n (t) =/braceleftbigg F1(t),x∈(−1,t0) F2(t),x∈(t0,1)/bracerightbigg , (2. 32) where F1(t) =∞/summationdisplay n=0cn(xnqn+gn)P(α+η,β−η) n (x), F2(t) =∞/summationdisplay n=0cn(xnrn+fn)P(α+η,β−η) n (x), ©200 1 CRC Press LLC and cn=Γ (n+β+ 1) Γ (n+β+ 1−η). After rescaling both known and unknown coeﬃcients via {yn,ˆfn,ˆgn}=Γ (n+ 1 +β) Γ (n+ 1 +β−η)/bracketleftBig h(α+η,β−η) n/bracketrightBig1 2{xn,fn,gn}, (2. 33) we obtain∞/summationdisplay n=0ynˆP(α+η,β−η) n (t) =/braceleftbiggG1(t), t∈(−1,t0) G2(t), t∈(t0,1)/bracerightbigg , (2. 34) where G1(t) =∞/summationdisplay n=0(ynqn+ ˆgn)ˆP(α+η,β−η) n (t), G2(t) =∞/summationdisplay n=0(ynrn+ˆfn)ˆP(α+η,β−η) n (t). From this, we ﬁnally obtain the i.s.l.a.e. of the second kind (1−qs)ys+∞/summationdisplay n=0yn(qn−rn)ˆQ(α+η,β−η) sn (t0) = ˆgs+∞/summationdisplay n=0/parenleftBig ˆfn−ˆgs/parenrightBig ˆQ(α+η,β−η) sn (t0),(2. 35) wheres= 0,1,2,...This i.s.l.a.e. possesses very similar properties to those of (2.23). It is not possible, in general, to solve the regularised systems (2 .23) or (2.35) explicitly in closed form, except for certain choices of qnandrn.Without loss of generality we may suppose that qn= 0.As an example, consider rn=−A n(n+α+β+ 1) for some constant A.Then (2.23) implies that Y(x) =∞/summationdisplay n=0ynˆP(α+η,β−η) sn (x) satisﬁes 1 w(x)d dx/bracketleftbigg/parenleftbig 1−x2/parenrightbig w(x)dY dx(x)/bracketrightbigg +AY(x) = 1 w(x)d dx/bracketleftBigg /parenleftbig 1−x2/parenrightbig w(x)dˆF dx(x)/bracketrightBigg , x∈(−1,t0), ©200 1 CRC Press LLC Y(x)=ˆG(x),x ∈(t0,1) where ˆF(x)=∞/summationdisplay n=0ˆfnˆP(α+η,β−η) sn (x), ˆG(x)=∞/summationdisplay n=0ˆgnˆP(α+η,β−η) sn (x). Thediﬀere ntialequationm aybesolve dtoyield,when x∈(−1,t0), Y(x)=∞/summationdisplay n=0n(n+α+β+1) n(n+α+β+1)−AˆfnˆP(α+η,β−η) sn (x)+CH 1(x)+DH 2(x), whereH1,H2areapairoflinearlyindependentsolution sof 1 w(x)d dx/bracketleftbigg/parenleftbig 1−x2/parenrightbig w(x)dY dx(x)/bracketrightbigg +AY(x)=0, andC,D areconstants .Theconstantsar eexplicitlydeterminedbyenforcing continuityof Yanditsderi vativeatthepoint t0,andtheexpansionc oeﬃcie nts ofYarethenexplicitlycalculated. Moregenerally,th esameargume ntcanbeapplie dwhen rn=A1 n(n+α+β+1)+A2 n2(n+α+β+1)2+...+Ar nr(n+α+β+1)r toproduc eadiﬀere ntialequationoforder2 rthatmaybesolve dprovide dthe correspondin ghomogeneou sdiﬀerentialequationissol ved.AgainYismade fullydeterminat ebyenforcingcontinuityon Yanditsﬁrst2 r−1derivatives att0. Thisidealie sbehindvariou smethod stoimprovetheco nvergenc eof(2.23) undertruncation,byreplacingi twithamor erapidl yconvergentsystem.An exampleofthiste chniqu ewillbegiveninChapter4. Sometimes mixed-boundary value problems in potential theory or wave- scattering theory lead to dual series equations for which the parameter con- straints (namely α−η >−1,β >−1 on the pair (2 .4) and (2.5), orα > −1,β−η >−1 on the pair (2. 29) and (2. 30)) do not hold. We may over- come this diﬃculty by transforming the initial equations to an equivalent setwhich involve Jacobi polynomials with increased values of the indices. There are two ways to eﬀect such a transformation. One may apply the formula deduced from Rodrigues’ formula [59] for Jacobi polynomials: −2n(1−x) α(1 +x)βP(α,β) n (x) =d dx{(1−x)α+1(1 +x)β+1P(α+1,β+1) n−1 (x)}.(2. 36) ©200 1 CRC Press LLC Asecondwaysuccessivelyappliesth eintegrationformulae(1 .173)and(1.174). Withcompletelyarbitraryvaluesofth eparameters αorβ,thisconstruction israthe rcumbersome,s othatacompletelygeneralsolutionofthisproblem willnotbepresentedhere.Howe ver,wewilltreatspeciﬁcexamplesinthefol- lowin gsectionssolvin gEquation s(2.4)and(2.5)orEquation s(2.29)an d(2. 30),toillustrateth emeritsan dapplicabili tyoftheaboveme ntione dmeth ods. Thiscompletesou rexaminationofdualserie swithJacobipolynomialker- nels.Thefunction sˆQ(α,β) sn thatappearinth eﬁnalregularise dsystemplayan extremelyimportantrolebothinth eanalysisofan destablishin gthevalidity ofthesolution ,aswel lasawiderroleinth egenerali nvestigationofthesingle (ordouble )layerpote ntialdensity. 2.2Dualseriesequationsi nvolvingtrigonometricalfunc- tions Dualequationswithtrigonometrickernelsh avebee ninvestigatedbyagreat manyauthor s(see ,forexample,thebibliographyin[55]).Apparently,Tranter [62]wastheﬁrsttosolveequationsofthistypebyth edeﬁnitionmethod describedinSection1.4.Inthissectionweprese ntth eoriginalsolution, placing it in the context of the general theory developed in the previous section for dual series equations involving Jacobi polynomials P(α,β) n. The fundamental connection arises from the relationships (1 .151)–(1.154) between trigonometric functions and the Jacobi polynomials with indices α= ±1 2andβ=±1 2. In applications the parameter ηinvariably takes the value 1 2,so as noted at the end of the previous section, the case when α=β=−1 2 must be considered separately, since the solution described for the pair (2 .4) and (2.5) requires α−η >−1,whilst that for the pair (2 .29) and (2.30) requiresβ−η >−1; an initial transformation as described at the end of the previous section must be eﬀected. On the other hand, when α=β=1 2, the solution described in the previous section is valid. Let us consider the following dual series equations with kernels einϑ: bx0−g0+/summationdisplay n/negationslash=0{zn(1−qn)−ξn}einϑ= 0,|ϑ|<ϑ 0 (2. 37) ax0−f0+/summationdisplay n/negationslash=0|n| {zn(1−rn)−ζn}einϑ= 0,|ϑ|>ϑ 0 (2. 38) where the unknown coeﬃcient sequence {zn}n/negationslash=0will be assumed to lie in l2(1). The coeﬃcients a,b,g 0,f0and the sequence coeﬃcients ξn,ζn,qn,rn are assumed to be known; in addition, we suppose that q−n=qn,rn=r−n ©200 1 CRC Press LLC and lim |n|→∞qn= lim |n|→∞rn= 0. Introduce the following notation: /braceleftbigg xn yn/bracerightbigg =/braceleftbigg zn+z−n zn−z−n/bracerightbigg ;/braceleftbigg gn en/bracerightbigg =/braceleftbigg ξn+ξ−n ξn−ξ−n/bracerightbigg ;/braceleftbigg fn hn/bracerightbigg =/braceleftbigg ζn+ζ−n ζn−ζ−n/bracerightbigg . Then the pair of equations (2 .37) and (2.38) is equivalent to the two pairs of functional equations in which the unknowns xn,ynare decoupled: bx0−g0+∞/summationdisplay n=1{xn(1−qn)−gn}cosnϑ= 0,ϑ∈(0,ϑ0) (2. 39) ax0−f0+∞/summationdisplay n=1n{xn(1−rn)−fn}cosnϑ= 0,ϑ∈(ϑ0,π) (2. 40) and ∞/summationdisplay n=1{yn(1−qn)−en}sinnϑ= 0, ϑ∈(0,ϑ0) (2. 41) ∞/summationdisplay n=1n{yn(1−rn)−hn}sinnϑ= 0, ϑ∈(ϑ0,π). (2. 42) Let us consider ﬁrst the pair (2 .41) and (2.42) with sine function kernels; the pair with cosine function kernels will be treated later. Set z= cosϕ, z0= cosϕand use (1. 153) to obtain ∞/summationdisplay n=1n{An(1−rn)−dn}P(1 2,1 2) n−1(z) = 0, z∈(−1,z0) (2. 43) ∞/summationdisplay n=1{An(1−qn)−cn}P(1 2,1 2) n−1(z) = 0, z∈(z0,1) (2. 44) where {An,dn,cn}=√π 2Γ (n+ 1) Γ/parenleftbig n+1 2/parenrightbig{yn,hn,en}. (2. 45) The rescaled unknowns {An}∞ u=1lie inl2.Equations (2 .43), (2.44) are of the form (2.4), (2.5) because λn−1/parenleftbig1 2,1 2,1 2/parenrightbig =n,and we may conclude that (1−rs)ˆAs+∞/summationdisplay n=1ˆAn(rn−gn)ˆQ(0,1) n−1,s−1(z0) =ˆds+∞/summationdisplay n=1/parenleftBig ˆcn−ˆdn/parenrightBig ˆQ(0,1) n−1,s−1(z0),(2. 46) ©200 1 CRC Press LLC wheres=1,2,...,and {ˆAs,ˆds,ˆcs}=/radicalbigg 2 nΓ/parenleftbig n+1 2/parenrightbig Γ(n){An,ds,cs}=/radicalbiggnπ 2{yn,hn,en}. Notic ethat{ˆAs}∞ s=1∈l2. WenowturntoEquation s(2.39)and(2.40)withcosin efunctio nkernels; aninitia ltransformatio noftheparamete rvaluesisneeded.First,replace thecosin etermsbytheirJacob ipolynomia lreprese ntation(1.151).Then integrat ebothsidesoftheseequation susingFormula(2.36).(Thisterm- by- termintegratio nisjustiﬁe dinthesamewayasinthepreviou ssection ,using result sinAppendixD.2.)Wethenintegrat eusingFormula(1.174)toobtain (1+t)3 2∞/summationdisplay n=1Γ(n+1) Γ/parenleftbig n+1 2/parenrightbig{xn(1−rn)−fn}P(−1 2,3 2) n−1(t)= 2√π(ax0−f0){4(1+t)1 2−(1−t)1 2[π+2arcsint]},t∈(−1,t0),(2.47) (1−t)1 2∞/summationdisplay n=1Γ(n) Γ/parenleftbig n+1 2/parenrightbig{xn(1−qn)−gn}P(1 2,1 2) n−1(t)= −2√π(bx0−g0)(1+t)−1 2/braceleftBigπ 2−arcsint/bracerightBig ,t∈(t0,1),(2.48) wheret=cosθ,t0=cosθ0. Followingthestandar dschemedescri bedinSectio n2.1,weobtai nthedual series equations (1 +t)∞/summationdisplay n=1{xn(1−rn)−fn}P(0,1) n−1(t) =−2 (ax0−f0) ln[1 2(1−t)], t∈(−1,t0),(2. 49) (1 +t)∞/summationdisplay n=1{xn(1−qn)−gn}P(0,1) n−1(t) =−2 (bx0−g0), t∈(t0,1),(2. 50) where the unknowns {xn}∞ n=1∈l2(1). The following deﬁnite integral /integraldisplayz −1π 2+ arcsinx√1−x√z−xdx=−πln1−z 2(2. 51) which occurs in this process may be evaluated from the transform −√ 2πln/parenleftbigg cosφ 2/parenrightbigg =/integraldisplayφ 0θsin1 2θdθ√cosθ−cosφ. (2. 52) ©200 1 CRC Press LLC Introducing new coeﬃcients {ˆxn,ˆfn,ˆgn}=/radicalbigg 2 n{xn,fn,gn} (2. 53) we transform (2 .49) and (2.50) to F(t) =/braceleftbigg F1(t), t∈(−1,t0) F2(t),t∈(t0,1)(2.54) where F(t) = (1 +t)∞/summationdisplay n=1ˆxnˆP(0,1) n−1(t), (2.55) F1(t) =−2 (ax0−f0) ln/bracketleftbigg1 2(1−t)/bracketrightbigg + (1 +t)∞/summationdisplay n=1/parenleftBig ˆxnrn+ˆfn/parenrightBig ˆP(0,1) n−1(t), (2. 56) and F2(t) =−2 (bx0−g0) + (1 +t)∞/summationdisplay n=1(ˆxnqn+ ˆgn)ˆP(0,1) n−1(t). (2. 57) The rescaled solution {ˆxn}∞ n=1belongs tol2(2). Multiplying both sides of (2. 54) by ˆP(0,1) m−1(t)andintegratin gover[−1,1],andemployingthepropertiesof theincomplet escalarproduct(seeAppendix(B.6)) ,weobtain (1−rm) ˆxm−∞/summationdisplay n=1{ˆxn(qn−rn) + ˆgn−ˆfn}ˆQ(0,1) n−1,m−1(t0) = ˆfm+ 2x0/braceleftbigg −1−t0 mˆP(1,0) m−1(t0)/bracketleftbigg b−aln/parenleftbigg1−t0 2/parenrightbigg/bracketrightbigg +a1 +t0 m2ˆP(0,1) m−1(t0)/bracerightbigg + 2/braceleftbigg1−t0 mˆP(1,0) m−1(t0)/bracketleftbigg g0−f0ln/parenleftbigg1−t0 2/parenrightbigg/bracketrightbigg −f01 +t0 m2ˆP(0,1) m−1(t0)/bracerightbigg , (2. 58) wherem= 1,2,.... Whatever the value of the constant x0,the solution {ˆxm}∞ m=1of the system (2. 58) lies in l2; however, the value x0must be chosen so that it also lies in l2(2). This depends upon the smoothness of the function F,which is related to the rate of decrease of its Fourier coeﬃcients [49, 79]. Fis continuous everywhere on the interval [ −1,1] because (2. 55) is a uniformly convergent series. The functions F1andF2are continuous on the sub-intervals [ −1,t0) and (t0,1] respectively, so the only point where the function Fmay lose continuity is att0; observing that Fis continuous at this point gives an equation for the constantx0, namely, F1(t0) =F2(t0). (2. 59) ©200 1 CRC Press LLC From this condition we ﬁnd x0=c/bracketleftbigg g0−f0ln/parenleftbigg1−t0 2/parenrightbigg/bracketrightbigg + 1 +t0 2c∞/summationdisplay n=1{ˆxn(qn−rn) + ˆgn−ˆfn}ˆP(0,1) n−1(t0).(2. 60) where c=/bracketleftbigg b−aln/parenleftbigg1−t0 2/parenrightbigg/bracketrightbigg−1 . Combined with (2 .58),the relationship (2 .60) gives the solution of the dual series equations involving trigonometric functions cos nϑ. Let us substitute the expression (2 .60) forx0in Equation (2 .58), keeping in mind the relation- ship(seeAppendix ,(B.171)) ˆQ(0,1) n−1,m−1(t0) =(1−t0)2 mˆP(0,1) n−1(t0)ˆP(1,0) m−1(t0) +n mˆQ(1,0) n−1,m−1(t0).(2. 61) As a result we obtain (1−rm)Xm−∞/summationdisplay n=1{Xn(qn−rn) +Gn−Fn}× /braceleftBigg ˆQ(1,0) n−1,m−1(t0) +a(1 +t0)2 b−aln/parenleftbig1 2(1−t0)/parenrightbigˆP(0,1) n−1(t0) nˆP(0,1) m−1(t0) m/bracerightBigg =Fm+ 2ag0−f0b b−aln/parenleftbig1 2(1−t0)/parenrightbig1 +t0 mˆP(0,1) m−1(t0) (2. 62) wherem= 1,2,..., and {Xm,Gm,Fm}=m{ˆxm,ˆgm,ˆfm}. (2. 63) Because {Xm}∞ m=1lies inl2,the solution {ˆxm}lies inl2(2) as required. This completes the regularisation of the dual series (2 .39) and (2.40) or (2.41) and (2.42),and hence of the original system (2 .37) and (2.38).There is a companion set of dual series, in which the sub-intervals on which the indi- vidual equations are interchanged. It is easily shown that they reduce to the same equations as (2 .39) and (2.40) or (2.41) and (2.42) via the replacements t0→ −t1(ϑ1=π−ϑ0, t1= cosϑ1=−cosϑ0=−t0),ˆAs→(−1)sˆAs, {Xm,Gm,Fm} →(−1)m{Xm,Gm,Fm}. To complete our consideration of dual series equations involving trigono- metric kernels, we now consider the pairs of functional equations /braceleftbigg/summationtext∞ n=0{xn(1−qn)−gn}cos/parenleftbig n+1 2/parenrightbig ϑ= 0, ϑ ∈(0,ϑ0)/summationtext∞ n=0/parenleftbig n+1 2/parenrightbig {xn(1−rn)−fn}cos/parenleftbig n+1 2/parenrightbig ϑ= 0,ϑ∈(ϑ0,π)(2.64) ©200 1 CRC Press LLC and /braceleftbigg/summationtext∞ n=0{yn(1−qn)−en}sin/parenleftbig n+1 2/parenrightbig ϑ=0,ϑ ∈(0,ϑ0)/summationtext∞n=0/parenleftbig n+1 2/parenrightbig {yn(1−rn)−hn}sin/parenleftbig n+1 2/parenrightbig ϑ=0,ϑ∈(ϑ0,π).(2.65) Inaddition ,weconsiderthecompanionequation sinwhichth esub-inter vals ofdeﬁnitionoftheseequationsar einterchanged: /braceleftbigg/summationtext∞n=0/parenleftbig n+1 2/parenrightbig {xn(1−rn)−fn}cos/parenleftbig n+1 2/parenrightbig ϑ=0,ϑ∈(0,ϑ0)/summationtext∞n=0{xn(1−qn)−gn}cos/parenleftbig n+1 2/parenrightbig ϑ=0,ϑ ∈(ϑ0,π)(2.66) and /braceleftbigg/summationtext∞n=0/parenleftbig n+1 2/parenrightbig {yn(1−rn)−hn}sin/parenleftbig n+1 2/parenrightbig ϑ=0,ϑ∈(0,ϑ0)/summationtext∞n=0{yn(1−qn)−en}sin/parenleftbig n+1 2/parenrightbig ϑ=0,ϑ ∈(ϑ0,π).(2.67) However,fromtheelementaryrelationships cos/parenleftbig n+1 2/parenrightbig (π−θ)=(−1)nsin/parenleftbig n+1 2/parenrightbig θ, sin/parenleftbig n+1 2/parenrightbig (π−θ)=(−1)ncos/parenleftbig n+1 2/parenrightbig θ, itisevide ntthatthepair(2.67)isofth esametypeasth epair(2.64),and alsothatth epair(2.66)isofthesam etypeas(2.65).T hus,weshallconsider onlythepair s(2.64)and(2.65)andﬁn dsolution swith {xn,yn}∞ n=0∈l2(1). Usingtheidentities(1 .152)an d(1.154),andsetting t=cosϑ,t0=cosϑ0, werefor mulatetheseequationsi ntermsofJacobipolynomialsas ∞/summationdisplay n=0/parenleftbig n+1 2/parenrightbig Γ(n+1) Γ/parenleftbig n+1 2/parenrightbig{xn(1−rn)−fn}P(−1 2,1 2) n (t)=0,t∈(−1,t0), (2.68) ∞/summationdisplay n=0Γ(n+1) Γ/parenleftbig n+1 2/parenrightbig{xn(1−qn)−gn}P(−1 2,1 2) n (t)=0,t∈(t0,1)(2.69) and ∞/summationdisplay n=0/parenleftbig n+1 2/parenrightbig Γ(n+1) Γ/parenleftbig n+1 2/parenrightbig{yn(1−rn)−hn}P(1 2,−1 2) n (t)=0,t∈(−1,t0), (2.70) ∞/summationdisplay n=0Γ(n+1) Γ/parenleftbig n+1 2/parenrightbig{yn(1−qn)−en}P(1 2,−1 2) n (t)=0,t∈(t0,1). (2.71) Thegeneraltheory,developedinSection2.1,isapplicabletoth esecond pair of dual equations, (2 .70) and (2.71). We set η=α=1 2,β=−1 2,so that λn(α,β;η) =n+1 2, and then represent these equations in the standard form ∞/summationdisplay n=0λn/parenleftbigg1 2,−1 2;1 2/parenrightbigg {y∗ n(1−rn)−h∗ n}P(1 2,−1 2) n (t) = 0, t∈(−1,t0) (2. 72) ©200 1 CRC Press LLC ∞/summationdisplay n=0{y∗ n(1−qn)−e∗ n}P(1 2,−1 2) n (t) = 0, t∈(t0,1) (2. 73) where {y∗ n,h∗ n,e∗n}=Γ (n+ 1) Γ/parenleftbig n+1 2/parenrightbig{yn,hn,en}. (2. 74) The regularised system from (2. 65) is thus directly obtained from (2 .23): the rescaled coeﬃcients and unknowns {Ys,Hs,Es}=/parenleftbigg s+1 2/parenrightbigg1 2 {ys,hs,es} (2. 75) satisfy (1−rs)Ys+∞/summationdisplay n=0Yn(rn−qn)ˆQ(0,0) sn(t0) =Hs+∞/summationdisplay n=0(Es−Hs)ˆQ(0,0) sn(t0) (2. 76) wheres= 0,1,2,.... Notice that in this case the incomplete inner product ˆQ(0,0) snis simply an incomplete inner product of normalised Legendre polyno- mials ˆPn=ˆP(0,0) n=/parenleftbig n+1 2/parenrightbig1 2Pn: ˆQ(0,0) sn(t0) =/integraldisplay1 t0ˆPs(t)ˆPn(t)dt. (2. 77) Let us now consider the remaining dual series equations, (2 .64). Instead of applying the variant (2. 36) of Rodrigues’ formula as was done previously (cf. (2. 47),(2. 48)), we apply the integration Formulae (1 .173) and (1 .174). First we use the relationship (1 .173) withα=−1 2,β=1 2, /integraldisplay1 t(1−x)−1 2P(−1 2,1 2) n (x)dx=(1−t)1 2 n+1 2P(1 2,−1 2) n (t) (2. 78) and integrate both parts of Equations (2 .68) and (2.69).(The term-by-term integration of a square-summable Fourier series is justiﬁed.) As a result, we obtain ∞/summationdisplay n=0Γ (n+ 1) Γ/parenleftbig n+1 2/parenrightbig{xn(1−rn)−fn}P(1 2,−1 2) n (t) =C(1−t)−1 2, t∈(−1,t0), (2. 79) ∞/summationdisplay n=0Γ (n+ 1) Γ/parenleftbig n+3 2/parenrightbig{xn(1−qn)−gn}P(1 2,−1 2) n (t) = 0, t∈(−1,t0),(2. 80) whereCis a constant to be determined later. This is in standard form for the application of the Abel integral transform method outlined in the previous ©200 1 CRC Press LLC section (with α=η=1 2,β=−1 2). The ﬁrst step is to integrate (2 .79) again, but using Formula (1 .174) withα=1 2,β=−1 2: /integraldisplayt −1(1 +x)−1 2P(1 2,−1 2) n (x)dx=(1 +t)1 2 n+1 2P(−1 2,1 2) n (t). (2. 81) We ﬁnd (1 +t)1 2∞/summationdisplay n=0Γ (n+ 1) Γ/parenleftbig n+3 2/parenrightbig{xn(1−rn)−fn}P(−1 2,1 2) n (t) =C/parenleftBigπ 2+ arcsint/parenrightBig , t∈(−1,t0).(2. 82) Repeating the steps of the method described in the previous section converts Equations (2 .82) and(2.80) to the equivalent pair ∞/summationdisplay n=0xnPn(t) =/braceleftbiggF1(t), t∈(−1,t0) F2(t), t∈(t0,1)/bracerightbigg , (2. 83) where F1(t) =/parenleftbigg2 π/parenrightbigg1 2 K/parenleftBigg/radicalbigg 1 +t 2/parenrightBigg C+∞/summationdisplay n=0(xnrn+fn)Pn(t), F2(t) =∞/summationdisplay n=0(xnqn+gn)Pn(t), andKdenote sthecomplet eellipti cintegra lofﬁrstkind(seeAppendix ,(B. 78)). The value of the constant is determined by C=/parenleftBigπ 2/parenrightBig1 2/braceleftBigg K/parenleftBigg/radicalbigg 1 +t0 2/parenrightBigg/bracerightBigg−1∞/summationdisplay n=0{xn(qn−rn) +gn−fn}P(0,0) n(t0) (2. 84) and the coeﬃcients {xn}∞ n=0satisfy (1−rs)xs−∞/summationdisplay n=1{xn(qn−rn)Q(0,0) ns(t0) =/parenleftbigg2 π/parenrightbigg1 2 C/integraldisplayt0 −1K/parenleftBig/radicalbig (1 +t)/2/parenrightBig Pn(t)dt +fs+∞/summationdisplay n=1(gn−fn)Q(0,0) ns(t0),(2. 85) wheres= 0,1,2,.... Note that Q(0,0) nsis the unnormalised incomplete scalar product. The integral appearing in (2. 85) may be simply expressed in terms of complete elliptic integrals (see later, (5. 50)). ©200 1 CRC Press LLC Thiscompletesthesolutionofthedualseries(2.64)andconclude sour regularisationofdualserie sequation swit hvariou stypesoftrigonometric kernels. 2.3Dualseriesequationsinvolvingass ociatedLegendre functions Theass ociatedLegendrefunctions Pm nprovid eanotheri nterestin gands pe- cialsetofkernelsfordualserie sequations,wort hyofexaminationintheirown right.Because Pm nisessentiall ythemthderivati veoftheLegendr epolynomial Pn,m-foldintegrationofth edualserie sequationsimmediatel yproduce sdual seriesequationswithLegendrepolynomialkernel sthatar ereadil ysolvable. Thissectionexamine sthesolutionobtainedbythi ssimplepr ocess .Forlarge m,howe ver,theresultings chem eisnumericallyunstable;twostablem odiﬁ- cationsarethereforedescri bed.Th eadvantagesandlimitationsofth eoriginal andmodiﬁedsystemsarediscussed.Theseresultswereobtaine djointlywith Yu.A.Tuchkin ;someofthemap pearin[72]. Wethereforeconsiderdualseriesequation sinvolvingassociate dLegendre functionsPm n(cosθ),andexploitth esolutionalread yobtaine dinSection2.1. The indexmis a ﬁxed nonnegative integer. The dual series equations ∞/summationdisplay n=mxm n(1−εn)Pm n(cosθ) =G(θ), θ∈(0,θ0) (2. 86) ∞/summationdisplay n=m(2n+ 1)xmn(1−µn)Pm n(cosθ) =F(θ), θ∈(θ0,π) (2. 87) are to be solved for the unknown coeﬃcients {xm n}∞n=m. The quantities {εn}∞ n=0,{µn}∞n=0are assumed to be known sequences of, in general, complex quantities decreasing at least as fast as O/parenleftbig n−2/parenrightbig asn→ ∞ : εn=O/parenleftbig n−2/parenrightbig ;µn=O/parenleftbig n−2/parenrightbig . (2. 88) The functions G,F are assumed to be expandable in Fourier-Legendre series G(θ) =∞/summationdisplay n=mgm nPm n(cosθ) , F(θ) =∞/summationdisplay n=m(2n+ 1)fm nPm n(cosθ). (2. 89) ©200 1 CRC Press LLC wherethecoeﬃcie ntsgm nandfm nareoftheformgm n=αnn−mandfm n= βnn−m−1andsatisfy ∞/summationdisplay n=m|αn|2<∞;∞/summationdisplay n=m|βn|2<∞. (2.90) Furthermore ,alltheseriescontaine din(2.86),(2.87),and(2.89)areassumed tobetheFourie rseriesoftheirrespectivesums ,i.e.,areconvergentinthe weightedmeansquar esensewithweightwm,m(x)=/parenleftbig 1−x2/parenrightbigm. Whenn/greaterorequalslantm,therelationshi pbetweentheassociatedLegendr efunctions Pm nandtheJacob ipolynomial sP(m,m ) n−mis(seeAppendix ,(B.48)) Pm n(cosθ)=2−msinmθΓ(n+m+1) Γ(n+1)P(m,m ) n−m(cosθ). (2.91) Theconnectio nwithLegendr epolynomial sis Pm n(x)=(−1)m/parenleftbig 1−x2/parenrightbigm 2dm dxmPn(x). (2.92) Intermsoftheparameter softhedualequation sconsidere dinSectio n2.1, α=β=m,andη=1 2. Because of this connection, it is natural to seek the solution of the pair (2.86) and (2.87) in the class l2(2m) deﬁned by (2 .9): {xm n}∞n=m∈l2(2m). (2. 93) This condition which appears naturally in both potential theory and wave- scattering theory for open spherical surfaces, is equivalent to the boundedness condition for the energy integral, which is taken over a ﬁnite volume including the edges. Deﬁning Rn(x) =Pn+1(x)−Pn−1(x), the Legendre polynomials obey (see (1. 123)), (2n+ 1)Pn(x) =d dxRn(x). (2. 94) All series in (2 .86), (2.87) are (generalised) Fourier series, so they can be in- tegrate dterm- by-term(seeAppendixD.2).Setx=cosθ.Divid e(2.86)and (2.87) by/parenleftbig 1−x2/parenrightbigm 2; then integrate each equation m-times; a further integra- tion of (2.87) is made using (2. 94). As a result of this process, polynomials (inx) of degree m−1 andm,respectively appear on the right-hand sides of these equations with coeﬃcients deriving from integration constants. Express each polynomial as a ﬁnite sum in terms of Legendre polynomials, with some ©200 1 CRC Press LLC undetermined coeﬃcients to obtain ∞/summationdisplay n=mxm n(1−εn)Pn(cosθ) = m−1/summationdisplay n=0Cm nPn(cosθ) +∞/summationdisplay n=mfm nPn(cosθ), θ∈(0,θ0),(2. 95) ∞/summationdisplay n=mxm n(1−µu)Rn(cosθ) = m−1/summationdisplay n=0(Cm n+Amn)Rn(cosθ) +∞/summationdisplay n=mgm nRn(cosθ),θ∈(θ0,π) (2. 96) where coeﬃcients Am n,Cm ndenote arbitrary constants of integration. This system has a solution in l2, i.e., {xm n}∞n=m∈l2(0)≡l2. (2. 97) Each solution of (2 .86) and (2.87) is a solution of (2 .95) and (2.96) whatever the values of the coeﬃcients Am n,Cm nmay be. However, any solution of (2 .95) and (2.96) depends on the 2 marbitrary constants Amn,Cm nand so in general is not a solution of (2 .86) and (2.87). We now show how to determine Am n, Cm nso that (2.93) is satisﬁed; the solution of (2 .95) and (2.96) will also then be the solution of (2 .86) and (2.87). This depends upon diﬀerentiating (2 .95) and (2.96) the required number of times. We now use the Dirichlet-Mehler integral representations for Legendre poly- nomials (1.149) and (1 .150) and observe that Rn(cosθ) =−2√ 2 π/integraldisplayθ 0sin/parenleftbig n+1 2/parenrightbig ϕsinϕ (cosϕ−cosθ)1 2dϕ =2√ 2 π/integraldisplayπ θcos/parenleftbig n+1 2/parenrightbig ϕsinϕ (cosϕ−cosθ)1 2dϕ. (2. 98) Transfer all terms in (2 .95) and (2.96) to the left-hand sides of these equa- tions, use the Dirichlet-Mehler integrals, and substitute the expression (2 .98). Invert the order of summation and integration (the validity of this operation is ensured by (2 .88) and (2.97)) to obtain two integral relationships, each of which is a homogenous Abel integral equation with a unique zero solution. As a result we obtain the following dual series equations. ∞/summationdisplay n=mxm ncos/parenleftbigg n+1 2/parenrightbigg θ= m−1/summationdisplay n=0Cm ncos/parenleftbigg n+1 2/parenrightbigg θ+/braceleftbiggF1(θ), θ∈(0,θ0) F2(θ), θ∈(θ0,π)/bracerightbigg ,(2. 99) ©200 1 CRC Press LLC where F1(θ) =∞/summationdisplay n=m(xm nεn+fm n) cos/parenleftbigg n+1 2/parenrightbigg θ, F2(θ) =∞/summationdisplay n=m(xm nµn+gm n) cos/parenleftbigg n+1 2/parenrightbigg θ+m−1/summationdisplay n=0Am ncos/parenleftbigg n+1 2/parenrightbigg θ. The set {cos/parenleftbig n+1 2/parenrightbig θ}∞ n=0is orthogonal, so multiplying both sides of Equa- tion (2.99) by cos/parenleftbig s+1 2/parenrightbig θand integrating term-by-term over [0 ,π], we ﬁnd Cm s+∞/summationdisplay n=m{xm n(εn−µn) + (fm n−gm n)}Qsn(θ0) =−∞/summationdisplay n=0Amn{δsn−Qsn(θ0)},(2. 100) wheres= 0,1,2,...,m −1,and xms(1−µs) +∞/summationdisplay n=m{xmn(µn−εn)Qsn(θ0) =gm s+∞/summationdisplay n=m(fm n−gm n)Qsn(θ0)−m−1/summationdisplay n=0AmnQsn(θ0),(2. 101) wheres=m,m + 1,...,andQsn(θ0) =ˆQ(−1 2,1 2) sn (cosθ0). Equation (2 .101) is an inﬁnite system of the linear algebraic equations of the second kind for the unknowns {xm n}∞n=m; its solution depends on the m constantsAm 0,...,Amm−1. Let us introduce the formal notation Dk m(ϑ) for thek-th derivative (with respect toϑ) of ∞/summationdisplay n=m{xm n(εn−µn) + (fm n−gm n)}cos/parenleftbigg n+1 2/parenrightbigg ϑ−m−1/summationdisplay n=0Am ncos/parenleftbigg n+1 2/parenrightbigg ϑ. (2. 102) Recollect our assumption that the solution of (2 .101) belongs to the class l2(2m). From standard results, which connect the smoothness of a function with the rate of decrease of its Fourier coeﬃcients [49, 79], the enforcement of the aggregate of mconditions Dk m(ϑ0) = 0, k = 0,1,2,...,m −1 (2. 103) on Equations (2 .99) is necessary and suﬃcient for the solution (2 .101) to belong to the class l2(2m). Assuming this, one can diﬀerentiate the Equations ©200 1 CRC Press LLC (2.102) term-by-term. Combining (2 .101) with (2 .103) (the result of term- by-term diﬀerentiation of (2 .102) at the point θ=θ0), we are led to an inﬁnite system of the linear algebraic equations for the aggregate of unknowns {Am n}m−1 n=0and{xmn}∞n=m∈l2(2m). It can be shown that Equations (2 .86) and (2.87) are equivalent to the set of Equations (2 .101)and (2.103); thus, we have successfully converted the original dual series equations (2 .86) and (2.87) to an inﬁnite system of linear algebraic equations, which can be solved by various numerical methods. The solution has asymptotic behaviour xm s=Dk m(θ0) /parenleftbig s+1 2/parenrightbigm+1.2 πΨm s(θ0) +O/parenleftbig s−m−2/parenrightbig , (2. 104) ass→ ∞,where Ψm s(θ0) = sin/parenleftbig s+1 2/parenrightbig θ0or cos/parenleftbig s+1 2/parenrightbig θ0according as mis even or odd. The simplicity in calculating the matrix elements of the system (2 .100), (2.101) and the condition (2 .103) is attractive: only trigonometric functions are used. However, it can be shown that for large mthis scheme is unstable, and leads to signiﬁcant errors in the calculation of the coeﬃcients Am n. But providedmis not large, this system is very suitable for numerical calculation. Let us therefore modify the system to improve its stability. Write (2 .103) as mequations for the unknown values Am n: /braceleftBigg dk dθk/bracketleftBiggm−1/summationdisplay n=0Am ncos/parenleftbigg n+1 2/parenrightbigg θ−∞/summationdisplay n=mWm ncos/parenleftbigg n+1 2/parenrightbigg θ/bracketrightBigg/bracerightBigg θ=θ0= 0, (2. 105) wherek= 0,1,2,...m−1,and Wm n=xm n(εn−µn) +fm n−gm n. (2. 106) Assuming that conditions (2 .103) are satisﬁed, we wish to obtain a nu- merically stable algorithm. Let us consider the orthonormal family of Jacobi polynomials ( n≥k, kﬁxed), ˆP(k−1 2,k+1 2) n−k(cosθ) =(−1)k √π/braceleftbigg(n−k)! (n+k)!/bracerightbigg1 2/parenleftbigg1 sinθd dθ/parenrightbiggk/bracketleftBigg cos/parenleftbig n+1 2/parenrightbig θ cos1 2θ/bracketrightBigg . (2. 107) The coeﬃcients Am nadmit the representation Am n=∞/summationdisplay j=mWm jαj nm, n= 0,1,2,...,m −1 (2. 108) ©200 1 CRC Press LLC where the coeﬃcients αj nm(j≥m) are solutions of the equations m−1/summationdisplay n=kαj nm/braceleftbigg(n+k)! (n−k)!/bracerightbigg1 2ˆP(k−1 2,k+1 2) n−k(cosθ) =/braceleftbigg(j+k)! (j−k)!/bracerightbigg1 2ˆP(k−1 2,k+1 2) j−k(cosθ),(2. 109) fork= 0,1,2,...,m −1.For every ﬁxed j,the matrix of the system of Equa- tions (2.109) is upper triangular, so the solution can be easily obtained by a recursive procedure. Now diﬀerentiate (2 .99)mtimes to obtain an equivalent system of linear algebraic equations. Accepting the representation (2 .108) for the coeﬃcients Am n, the ﬁnal system is ˆxm s(1−µs)−∞/summationdisplay n=mˆxmn(εn−µn)Wm sn(θ0) = ˆgm s+∞/summationdisplay n=m/parenleftBig ˆfm n−ˆgm n/parenrightBig Wm sn(θ0),(2. 110) wheres=m,m + 1,m+ 2,...and /braceleftBig ˆxm n,ˆfm n,ˆgm n/bracerightBig =/parenleftbigg n+1 2/parenrightbiggm {xm n,fm n,gm n}, (2. 111) Wm sn(θ0) =Um sn(θ0)−m−1/summationdisplay n=k/parenleftbigg j+1 2/parenrightbiggm αn jmUm sj(θ0), (2. 112) and Um sj(θ0) =1 π/bracketleftbiggsin (s−j)θ0 s−j+ (−1)msin (s+j+ 1)θ0 s+j+ 1/bracketrightbigg (2. 113) with the understanding/bracketleftbiggsinnθ0 n/bracketrightbigg n=0=θ0. Thus, the initial dual series Equations (2 .86) and (2.87), with associated Legendre function kernels, are transformed to the equivalent system of linear algebraic Equations (2 .110); it is a second-kind equation that is a completely continuous perturbation of the identity operator in l2. However it is signif- icantly more stable than (2. 101) and (2. 103), albeit at the cost of rathermore complicated coeﬃcients. Another stable form may be derived as follows.Using the relationship (2 .107), we represent (2 .99) in equivalent form ∞/summationdisplay n=mxm nˆP(−1 2,1 2) n (cosθ)−m−1/summationdisplay n=0Cm nˆP(−1 2,1 2) n (cosθ) =/braceleftbiggF1(θ), θ∈(0,θ0) F2(θ), θ∈(θ0,π), ©200 1 CRC Press LLC (2. 114) where F1(θ) =∞/summationdisplay n=m(xm nεn+fm n)ˆP(−1 2,1 2) n (cosθ), F2(θ) =∞/summationdisplay n=m(xm nµn+gm n)ˆP(−1 2,1 2) n (cosθ) +m−1/summationdisplay n=0Am nˆP(−1 2,1 2) n (cosθ). For these orthonormal Jacobi polynomials the following diﬀerentiation for- mula holds when k≤n, [58], dk dxkˆP(−1 2,1 2) n (x) = [(n+k)!/(n−k)!]1 2ˆP(k−1 2,k+1 2) n−k(x) ; (2. 115) thek-fold derivative vanishes when k >n. Introduce the new unknowns and coeﬃcients {ym n,Fm n,Gm n}=/braceleftbigg(n+m)! (n−m)!/bracerightbigg1 2 {xm n,fm n,gm n}. (2. 116) It follows from (2 .90), (2.93), and(2.111) that {ym n,Fm n,Gm n}∞ n=m∈l2=l2(0). (2. 117) Assuming that condition (2 .93) is valid, we may diﬀerentiate the Equation (2.114)mtimes term-by-term with respect to x= cosθ. Keeping in mind the relationship (2 .105), we ﬁnd (setting x0= cosθ0), ∞/summationdisplay n=mym nˆP(m−1 2,m+1 2) n−m (x) =/braceleftbigg F1(x), x∈(−1,x0) F2(x), x ∈(x0,1)/bracerightbigg (2. 118) where F1(x) =∞/summationdisplay n=m(ym nµn+Fm n)ˆP(m−1 2,m+1 2) n−m (x), F2(x) =∞/summationdisplay n=m(ym nεn+Gm n)ˆP(m−1 2,m+1 2) n−m (x). The polynomials ˆP(m−1 2,m+1 2) s are orthonormal on [ −1,1] with weight function w(x) = (1 −x)m−1 2(1 +x)m+1 2; multiplying (2 .118) bywˆP(m−1 2,m+1 2) s−m and ©200 1 CRC Press LLC integrating term-by-term over [ −1,1],we obtain the inﬁnite system of linear algebraic equations (1−εs)ym s+∞/summationdisplay n=mym n(εn−µn)ˆQ(m−1 2,m+1 2) s−m,n−m(x0) =Gm n+∞/summationdisplay n=m(Fm n−Gmn)ˆQ(m−1 2,m+1 2) s−m,n−m(x0),(2. 119) wheres=m+1,m+2,..., and the usual normalised incomplete inner product has been employed. Comparing (2 .111) with (2 .116) we have /braceleftBig ˆxm n,ˆfm n,ˆgm n/bracerightBig =km n{ym n,Fm n,Gmn} (2. 120) where km n=/parenleftbigg n+1 2/parenrightbiggm/bracketleftbigg(n−m)! (n+m)!/bracketrightbigg1 2 . (2. 121) Observe that km n→1 asn→ ∞.In addition, it can be shown that the following relationship holds: ˆQ(m−1 2,m+1 2) s−m,r−m(cosθ0) =km r(km n)−1Wm sr(θ0). (2. 122) Formula (2.122) can be used for calculations of ˆQ(m−1 2,m+1 2) s−m,r−m(cosθ0), employ- ing (2.112). The systems (2 .110) and (2.119) are practically identical, diﬀering only in the normalisation (2 .120). In summary, we have shown how to regularise the special class of dual se- ries Equations (2 .86) and (2.87) containing associated Legendre functions as kernels. The simplest approach essentially integrated the series equations to obtain dual series equations with Legendre polynomial kernels, together with constants of integration that are uniquely determined by some diﬀerentiabil-ity conditions. This produced (2 .100), (2.101), and (2 .103).The simplicity in calculating the matrix elements of this system is attractive: however, asalready noted, it is unstable for large mand leads to signiﬁcant errors in the calculation of the coeﬃcients A m n. But provided mis not large, this system is quite suitable for numerical calculation. In order to rectify this instability, the modiﬁed system (2 .110) was derived, and its normalised variant (2 .119). Both these systems are stable, but the algorithm for calculation of the matrixcoeﬃcients is rather more complicated. ©200 1 CRC Press LLC 2.4SymmetrictripleseriesequationsinvolvingJacobi polynomials Tripleseriesequationsprese ntanobviou sextensionan dgeneralisationof dualseriesequations.Inthissectionweconside rsymmetric tripleseries equations,thekernelsofwhichareJacob ipolynomials P(α,β) n.Withou ta signiﬁcantlossofgenerality,werestric tatte ntiontokernel sofmostusein subsequentchapters,th eultraspheri calpolynomial P(α,α) n;theparameter η thatoccurredinSection2. 1willbeﬁxedtobe1 2.Moreover ,theinterval[ −1,1] issubdividedintothre esubi ntervalsonwhichth ecorrespondingfunctional equationsareenforced,s othatthemiddlesubintervalissymmetricabout0. Thustheter msymmetricequations highlightstwodiﬀere ntaspects:equality oftheparameter sαandβ,andasymmetricsu bdivisionofth efullintervalof deﬁnition[ −1,1].Nonsymmetri csubdivisionswil lbedeferredtoSection2.7. Retainingallthenotationintr oduce dinSection2. 1weconsiderequation sof two types, Type A and Type B, being, respectively, the sets of triple equations ∞/summationdisplay n=0{xn(1−qn)−gn}P(α,α) n (t) = 0, t∈(−1,−t0), (2. 123) ∞/summationdisplay n=0λn(α,α;1 2){xn(1−rn)−fn}P(α,α) n (t) = 0, t∈(−t0,t0), (2. 124) ∞/summationdisplay n=0{xn(1−qn)−gn}P(α,α) n (t) = 0, t∈(t0,1), (2. 125) and ∞/summationdisplay n=0λn(α,α;1 2){xn(1−rn)−fn}P(α,α) n (t) = 0, t∈(−1,−t0), (2. 126) ∞/summationdisplay n=0{xn(1−qn)−gn}P(α,α) n (t) = 0, t∈(−t0,t0), (2. 127) ∞/summationdisplay n=0λn/parenleftbig α,α;1 2/parenrightbig {xn(1−rn)−fn}P(α,α) n (t) = 0, t∈(t0,1), (2. 128) ©200 1 CRC Press LLC where we recall from deﬁnition (2 .3),the coeﬃcient λn/parenleftbig α,α;1 2/parenrightbig =n+α+1 2. The solution {xn}∞ n=0is sought in the class l2; in addition, we assume that {fn}∞ n=0,{gn}∞n=0∈l2. 2.4.1 Type A triple series equations Usingthesymmetr ypropertyofJacob ipolynomial s(seeAppendix ,(B.26), withβ=α), P(α,α) n (−t) = (−1)nP(α,α) n (t) we may transform the Equations (2 .123)–(2.125) to two sets of dual series equations, for the odd ( l= 1) and even ( l= 0) unknown coeﬃcients, re- spectively; the interval of deﬁnition of the dual equations is halved. The coeﬃcients satisfy (for l= 0,1) ∞/summationdisplay n=0{x2n+l(1−q2n+l)−g2n+l}P(α,α) 2n+l(z) = 0, z∈(−1,−z0),(2. 129) ∞/summationdisplay n=0λ2n+l(α,α;1 2){x2n+l(1−r2n+l)−f2n+l}P(α,α) 2n+l(z) = 0, z∈(−z0,0), (2. 130) In itself, this transformation does not construct an eﬀective solution of equa- tions of Type A. The key step is to connect the ultraspherical polynomialswith Jacobi polynomials [58]: P (α,α) 2n+l(z) =Γ (n+ 1) Γ (2n+ 1 +l)Γ (2n+α+ 1 +l) Γ (n+α+ 1)zlP(α,l−1 2) n/parenleftbig 2z2−1/parenrightbig .(2. 131) This transforms the dual Equations (2 .129) and (2 .130), which are deﬁned on [−1,0], to another set of dual equations that are deﬁned on the complete interval [ −1,1]. Setting u= 2z2−1 andu0= 2z2 0−1, we obtain ∞/summationdisplay n=0/parenleftbigg n+α+l 2+1 4/parenrightbigg/braceleftbig x∗ 2n+l(1−r2n+l)−f∗ 2n+l/bracerightbig P(α,l−1 2) n (u) = 0, u∈(−1,u0) (2. 132) ∞/summationdisplay n=0/braceleftbig x∗ 2n+l(1−q2n+l)−g∗ 2n+l/bracerightbig P(α,l−1 2) n (u) = 0, u∈(u0,1) (2. 133) where the rescaled coeﬃcients are /braceleftbig x∗ 2n+l,f∗ 2n+l,g∗ 2n+l/bracerightbig =Γ (n+ 1) Γ (2n+α+ 1 +l) Γ (2n+ 1 +l) Γ (n+α+ 1){x2n+l,f2n+l,g2n+l}. (2. 134) ©200 1 CRC Press LLC Inordertoapplythemeth oddevelopedinSectio n2.1,rewrit ethedual equations as ∞/summationdisplay n=0/braceleftbig Λl n(1−r2n+l)X2n+l−F2n+l/bracerightbigˆP(α−1 2,l) n (u) = 0, u∈(−1,u0) (2. 135) ∞/summationdisplay n=0{(1−q2n+l)X2n+l−G2n+l}ˆP(α−1 2,l) n (u) = 0, u∈(u0,1) (2. 136) where X2n+l=Γ (n+α+ 1) Γ/parenleftbig n+α+1 2/parenrightbig/braceleftbigg h(α−1 2,l) n/bracerightbigg1 2 x∗ 2n+l, (2. 137) G2n+l=Γ (n+α+ 1) Γ/parenleftbig n+α+1 2/parenrightbig/braceleftbigg h(α−1 2,l) n/bracerightbigg1 2 g∗ 2n+l, (2. 138) F2n+l=/parenleftbigg n+α+l 2+1 4/parenrightbiggΓ/parenleftbig n+l+1 2/parenrightbig Γ (n+l+ 1)/braceleftbigg h(α−1 2,l) n/bracerightbigg1 2 f∗ 2n+l, (2. 139) and Λl n=/parenleftbigg n+α+l 2+1 4/parenrightbiggΓ/parenleftbig n+l+1 2/parenrightbig Γ (n+l+ 1)Γ/parenleftbig n+α+1 2/parenrightbig Γ (n+α+ 1). (2. 140) FromField’ sformulafortheratioofGamm afunction s(seeAppendix ,(B. 7)), we deduce Λl n= 1 +O/parenleftbig n−2/parenrightbig ,asn→ ∞, (2. 141) and introduce the asymptotically small parameterεl ndeﬁned by εl n= 1−Λln=O/parenleftbig n−2/parenrightbig . (2. 142) After some rearrangement (2 .135), (2.136) become ∞/summationdisplay n=0X2n+lˆP(α−1 2,l) n (u) =/braceleftbigg F1(u), u ∈(−1,u0) F2(u), u ∈(u0,1)/bracerightbigg (2. 143) where F1(u) =∞/summationdisplay n=0/braceleftbig/bracketleftbig r2n+l+εl n(1−r2n+l)/bracketrightbig X2n+l+F2n+l/bracerightbigˆP(α−1 2,l) n (u), F2(u) =∞/summationdisplay n=0{q2n+lX2n+l+G2n+l}ˆP(α−1 2,l) n (u). As usual, multiply both sides of Equation (2 .143) by the factor (1−u)α−1 2(1 +u)lˆP(α−1 2,l) n (u) ©200 1 CRC Press LLC andintegrateover[ −1,1].Th eresultisaninﬁnitesystemoflinearalgebraic equations,th ematrixoperatorofwhi chisacompletelyconti nuou sperturba- tionoftheidentity(i nl2): /braceleftbig 1−/bracketleftbig r2s+l+εl s(1−r2s+l)/bracketrightbig/bracerightbig X2s+l− ∞/summationdisplay n=0X2n+l/braceleftbig q2n+l−/bracketleftbig r2n+l+εl n(1−r2n+l)/bracketrightbig/bracerightbigˆQ(α−1 2,l) ns (u0) =F2s+l+∞/summationdisplay n=0(G2n+l−F2n+l)ˆQ(α−1 2,l) ns (u0),(2.144) wheres=0,1,2,....Thisregularise dsyste misvalidforbothe ven(l=0)or odd(l=1)coeﬃcients. Aremar kisinorder.Whe nα=1 2,thekernelsesse ntiallyreducetothe trigonometricfunctionssin nϑandεl n≡0foralln.However ,theprocedure aboveisapplicableonlywhe nα>−1 2.Tocircu mventth ediﬃcultyencoun- teredwhe nα=−1 2,(correspondin gtothekernel scosnθ)wemayus ethose devicesappliedtoobtainsolutionofsimilarequationsi npreviou ssections (basedonRodrigues’for mula,etc.). 2.4.2Ty peBtripleserie sequations Asimilarargume nttothatempl oyedi nthelastsectiontransformsth etriple series(2.126)–(2.128)totheanalogueof(2.132)and(2.133).Omittingthe preliminarystepsofthisdeduction,weobtai n(withth esam enotation) ∞/summationdisplay n=0/braceleftbig x∗ 2n+l(1−q2n+l)−g∗ 2n+l/bracerightbig P(α,l−1 2) n (u)=0,u∈(−1,u0)(2.145) ∞/summationdisplay n=0/parenleftbigg n+α+l 2+1 4/parenrightbigg/braceleftbig x∗ 2n+l(1−r2n+l)−f∗ 2n+l/bracerightbig P(α,l−1 2) n (u)=0, u∈(u0,1)(2.146) Theoddcase( l=1)ofth edualpair(2 .145),(2.146)issolvablebymeans ofthegeneraltheorydevelopedi nSection2.1,whe nα>−1.Weobtainthe regularised system (1−q2s+1)y2s+1+ ∞/summationdisplay n=0{q2n+1−[r2n+1+µn(1−r2n+1)]}y2n+1ˆQ(α+1 2,0) ns (u0) = ˆg2s+1+∞/summationdisplay n=0/parenleftBig ˆf2n+1−ˆg2n+1/parenrightBig ˆQ(α+1 2,0) ns (u0),(2. 147) ©200 1 CRC Press LLC wheres=0,1,2,...,and y2n+1=Γ/parenleftbig n+3 2/parenrightbig Γ(n+1)/braceleftbigg h(α+1 2,0) n/bracerightbigg1 2 x∗ 2n+1, (2.148) ˆg2n+1=Γ/parenleftbig n+3 2/parenrightbig Γ(n+1)/braceleftbigg h(α+1 2,0) n/bracerightbigg1 2 g∗ 2n+1, ˆf2n+1=Γ(n+α+1) Γ/parenleftbig n+α+3 2/parenrightbig/parenleftbigg n+α 2+3 4/parenrightbigg/braceleftbigg h(α+1 2,0) n/bracerightbigg1 2 f∗ 2n+1, and µn=1−/parenleftbigg n+α 2+3 4/parenrightbiggΓ(n+1)Γ(n+α+1) Γ/parenleftbig n+3 2/parenrightbig Γ/parenleftbig n+α+3 2/parenrightbig. (2.149) Theparamete rµnisasymptoticallysmall :µn=O/parenleftbig n−2/parenrightbig asn→∞. Intheevencas e(l=0)th eparameter sfalloutsid etherangeofapplicability ofth emethoddescribedinSection2.1.Thisnecessitatesth eapplicationof anothermethodthatwasuse dinth eanalysisofEquation s(2.39)and(2.40), whichca nbeconsidere dasaparticula rcaseofthemor egeneralEquations (2.145)and(2 .146)withvalues α=−1 2,l=0.Althoughthesolutioncanbe obtainedinthismor egeneralcase,weomitthedetails,an dconﬁneatte ntion toas peciﬁ cexamplethatwillbetreatedinSection4.4. 2.5 Relationships between series and integral equations This purpose of this section is to explain the relationship between some classes of series and integral equations, and to show how the scope of the Abel integral equation method may be expanded to establish such connec- tions. Dual integral equations will be considered in their own right in thenext section. The results of this section are based upon those obtained by W.E. Williams [76], [77]; A. A. Ashour [3]; and J. S. Lowndes [37]. Letmbe a ﬁxed nonnegative integer. We consider two basic kinds of dual series equations. The kernel of the ﬁrst employs associated Legendre functions ∞/summationdisplay n=mam nPm n(cosθ) =Fm(θ), θ∈(0,θ0), (2. 150) ∞/summationdisplay n=m(2n+ 1)am nPm n(cosθ) =Gm(θ), θ∈(θ0,π), (2. 151) ©200 1 CRC Press LLC whilst the second employs trigonometric kernels, ∞/summationdisplay n=1bnsinnϕ=f(ϕ), ϕ∈(0,ϕ0), (2. 152) ∞/summationdisplay n=1nbnsinnϕ=g(ϕ), ϕ∈(ϕ0,π). (2. 153) In addition, we consider two types of dual integral equations. The kernel of the ﬁrst is a Bessel function of integer order m, /integraldisplay∞ 0Am(λ)Jm(λρ)dλ=Em(ρ),0≤ρ<a (2. 154) /integraldisplay∞ 0λAm(λ)Jm(λρ)dλ=Hm(ρ), ρ>a (2. 155) whilst the second has a trigonometric kernel, /integraldisplay∞ 0B(µ) sin (µx)dµ=e(x),0≤x<b (2. 156) /integraldisplay∞ 0µB(µ) sin (µx)dµ=h(x), x>b (2. 157) The functions Fm,Gm,f,g,E m,Hm,eandhoccurring on the right-hand sides of (2.150)–(2.157) are assumed to be known; the equations are to be solved for the unknown coeﬃcients am n,bnand functions Am,B, respectively. Let us extend the domain of deﬁnition of the functions occurring in Equations (2.151) and (2 .155) in the following way. Let ∞/summationdisplay n=m(2n+ 1)am nPm n(cosθ) =/braceleftbiggCm(θ), θ ∈(0,θ0) Gm(θ), θ ∈(θ0,π)/bracerightbigg (2. 158) and/integraldisplay∞ 0λAm(λ)Jm(λρ)dλ=/braceleftbigg Lm(ρ),0≤ρ<a Hm(ρ), ρ>a/bracerightbigg . (2. 159) The relationship between the coeﬃcients am nandCm(θ),or between the co- eﬃcientsAmandLm(ρ), is found using the orthogonality of associated Leg- endre’s functions Pm non [0,π],or by using the Fourier-Bessel transform as appropriate: am n=1 2(n−m)! (n+m)!/integraldisplayθ0 0dθsinθCm(θ)Pm n(cosθ) + 1 2(n−m)! (n+m)!/integraldisplayπ θ0dθsinθGm(θ)Pm n(cosθ),(2. 160) ©200 1 CRC Press LLC Am(λ) =/integraldisplaya 0rLm(r)Jm(λr)dr+/integraldisplay∞ arHm(r)Jm(λr)dr. (2. 161) Now substitute these expressions for am norAmin (2.150) and (2 .154). This leads to two ﬁrst-kind Fredholm integral equations involving the unknown functionsCmandLm: /integraldisplayθ0 0dϑsinϑCm(ϑ)K1(ϑ,θ) = 2Fm(θ)−G∗ m(θ), θ∈(0,θ0),(2. 162) /integraldisplaya 0dr.rL m(r)K2(r,ρ) =E(ρ)−H∗(ρ),0≤ρ<a, (2. 163) where G∗m(θ) =1 πcotmθ 2/integraldisplayθ 0tan2m1 2ϕdϕ (cosϕ−cosθ)1 2/integraldisplayπ θ0Gm(ϑ) cotm1 2ϑ (cosϕ−cosϑ)1 2dϑ, (2. 164) H∗ m(ρ) =2 πρ−m/integraldisplayρ 0dzz2m (ρ2−z2)1 2/integraldisplay∞ adrrH(r) (r2−z2)1 2, (2. 165) and the kernels of these integral equations are K1(ϑ,θ) =∞/summationdisplay n=m(n−m)! (n+m)!Pm n(cosϑ)Pm n(cosθ), (2. 166) K2(r,ρ) =/integraldisplay∞ 0Jm(λr)Jm(λρ)dλ. (2. 167) These kernels admit the representation K1(ϑ,θ) =1 πcotmθ 2cotmϑ 2/integraldisplaymin(θ,ϑ) 0tan2m1 2ϕdϕ (cosϕ−cosθ)1 2(cosϕ−cosϑ)1 2, (2. 168) K2(r,ρ) =2 πr−mρ−m/integraldisplaymin(r,ρ) 0z2mdz (r2−z2)1 2(ρ2−z2)1 2. (2. 169) With the change of variables z= tanϕ 2, r= tanϑ 2, ρ= tanθ 2, it can be shown that K2/parenleftbigg tanϑ 2,tanθ 2/parenrightbigg = 2 cosθ 2cosϑ 2K1(ϑ,θ), (2. 170) ©200 1 CRC Press LLC so establishing a relationship between Equations (2 .162) and (2 .163); they are identical provided Lm/parenleftbigg tanϑ 2/parenrightbigg = cos3ϑ 2Cm(ϑ), (2. 171) Em/parenleftbigg tanϑ 2/parenrightbigg = cosϑ 2Fm(ϑ), (2. 172) H/parenleftbigg tanϑ 2/parenrightbigg = cos3ϑ 2Gm(ϑ), (2. 173) H∗/parenleftbigg tanϑ 2/parenrightbigg = cosϑ 2G∗ m(ϑ). (2. 174) Thus, we have demonstrated a one-to-one correspondence between the dual series Equations (2 .150) and (2 .151) and the dual integral Equations (2 .154) and (2.155), and their solutions. If the condition (2. 172) holds, we ﬁnd ∞/summationdisplay n=mam nPm n(cosθ) = secθ 2/integraldisplay∞ 0Am(λ)Jm/parenleftbigg λtanθ 2/parenrightbigg dλ. (2. 175) In a similar way, if the condition (2. 173) holds, we ﬁnd ∞/summationdisplay n=m(2n+ 1)am nPm n(cosθ) = sec3θ 2/integraldisplay∞ 0λAm(λ)Jm/parenleftbigg λtanθ 2/parenrightbigg dλ.(2. 176) We may now determine the relationship between solutions of these equa- tions. Multiply both parts of equations (2 .176) by the factor sin θPm k(cosθ) and integrate over [0 ,π], to ﬁnd am n= 2(n−m)! (n+m)!× /integraldisplay∞ 0/braceleftbigg/integraldisplay∞ 0λAm(λ)Jm(λu)dλ/bracerightbiggu√ 1 +u2Pm n/parenleftbigg1−u2 1 +u2/parenrightbigg du. (2. 177) On the other hand, using the Hankel transform, multiply both parts of (2 .176) by cos1 2θtan1 2θJm/parenleftbig µtan1 2θ/parenrightbig , and integrate with respect to ρ= tan1 2θover (0,∞). This gives the relation Am(λ) =1 2√ 2/integraldisplay1 −1/braceleftBigg∞/summationdisplay n=m(2n+ 1)am nPm n(x)/bracerightBigg Jm/parenleftBigg λ/radicalbigg 1−x 1 +x/parenrightBigg dx√1 +x. (2. 178) Thus the solution of dual- (or multiple-) series equations has its counterpart in the solution of the corresponding dual- (or multiple-) integral equations, and vice versa. ©200 1 CRC Press LLC Let us now demonstrate that the same is true for the pairs of Equations (2.152) and (2 .153) and (2 .156) and (2 .157). These equations are reducible to ﬁrst-kind Fredholm integral equations of the form /integraldisplayϕ0 0C(β)K3(β,ϕ)dϕ=π 2[f(ϕ)−g∗(ϕ)], ϕ∈(0,ϕ0), (2. 179) and/integraldisplayb 0l(y)K4(x,y)dy=π 2[e(x)−h∗(x)],0≤x<b, (2. 180) where h∗(x) =/integraldisplayx 0dtt (x2−t2)1 2/integraldisplay∞ bdyh(y) (y2−t2)1 2, (2. 181) g∗(ϕ) =1 2cosϕ 2/integraldisplayϕ 0dαsinα cos21 2α(cosα−cosϕ)1 2/integraldisplayπ ϕ0dβg(β) cos1 2β (cosα−cosβ)1 2,(2. 182) and the kernels of the integral equations are, respectively K3(β,ϕ) =∞/summationdisplay n=1sinnβsinnϕ n=1 2ln/vextendsingle/vextendsingle/vextendsingle/vextendsingletan 1 2ϕ+ tan1 2β tan1 2ϕ−tan1 2β/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (2. 183) K 4(x,y) =/integraldisplay∞ 0sinµxsinµy µdµ=1 2ln/vextendsingle/vextendsingle/vextendsingle/vextendsinglex+y x−y/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (2. 184) The kernels K 3(β,ϕ),K4(x,y) have a representation of the same form as (2.168) and (2 .169). A more general representation for this type of kernel is derived later in this section. The relationship between the integral Equations (2.179) and (2 .180) is established by observing that under the substitution x= tan1 2ϕ,y= tan1 2β, K4(x,y) =K3(β,ϕ). Thus, the integral equations are equivalent with the identiﬁcation l/parenleftbigg tanβ 2/parenrightbigg = 2 cos2β 2C(β), (2. 185) e/parenleftBig tanϕ 2/parenrightBig =f(ϕ), (2. 186) h∗/parenleftBig tanϕ 2/parenrightBig =g∗(ϕ), (2. 187) h/parenleftbigg tanβ 2/parenrightbigg = 2 cos2β 2g(β). (2. 188) Thus, if the following relation is valid ∞/summationdisplay k=1bnsinnϕ=/integraldisplay∞ 0B(µ) sin/parenleftBig µtanϕ 2/parenrightBig dµ, (2. 189) ©200 1 CRC Press LLC then so too is the relation ∞/summationdisplay k=1nbnsinnϕ= sec2ϕ 2/integraldisplay∞ 0µB(µ) sin/parenleftBig µtanϕ 2/parenrightBig dµ. (2. 190) Thus, the unknowns {bn}∞ n=1andBare connected by bn=2 πn/integraldisplayπ 0dϕsinnϕsec2ϕ 2/integraldisplay∞ 0µB(µ) sin/parenleftBig µtanϕ 2/parenrightBig dµ. (2. 191) The relationship stated above between some speciﬁc series and integral equations is not special and exists under more general conditions, which we now explore. The kernels of the series equations considered above are essen-tially Jacobi polynomials with symmetrical indices (see Formulae (1 .153) and (2. 91)): sinnϕ∝P( 1 2,1 2) n−1(cosϕ), Pm n(cosθ)∝P(m,m ) n−m(cosθ). On the other hand, since sin νx∝J1 2(νx),the corresponding integral Equa- tions (2. 154)–(2. 157) involve the Bessel functions of order equal to1 2or an integerm. We extend our considerations to series equations with ultraspher- ical polynomial kernels P(α,α) n,having arbitrary index α, and relate these to integral equations with Bessel function kernels of the same order α. So ﬁxing α, let us examine the extended class of dual equations ∞/summationdisplay n=0anP(α,α) n (x) =F(x), x∈(−1,x0), (2. 192) ∞/summationdisplay n=0λn(α,α;η)anP(α,α) n (x) =G(x), x∈(x0,1), (2. 193) and /integraldisplay∞ 0λ2ηA(λ)Jα(λρ)dλ=g(ρ),0≤ρ<1, (2. 194) /integraldisplay∞ 0A(λ)Jα(λρ)dλ=f(ρ), ρ> 1, (2. 195) where the parameter ηsatisﬁes 0 ≤η≤1 2,and the value λn(α,α;η) deﬁned by (2.3) has the property λn(α,α;η) =Γ (n+α+ 1 +η) Γ (n+α+ 1−η)=n2η/parenleftbig 1 +O(n−1)/parenrightbig asn→ ∞.(2. 196) Paralleling the argument previously employed, let us extend the domain of deﬁnition of the functions occurring in Equations (2 .193) and (2 .194), so that ∞/summationdisplay n=0λn(α,α;η)anP(α,α) n (x) =/braceleftbiggˆG(x), x∈(−1,x0) G(x), x∈(x0,1)/bracerightbigg (2. 197) ©200 1 CRC Press LLC and/integraldisplay∞ 0λ2ηA(λ)Jα(λρ)dλ=/braceleftbigg g(ρ),0≤ρ<1 ˆg(ρ), ρ> 1/bracerightbigg , (2. 198) where ˆGand ˆgare unknown functions to be determined. Using the orthogonal- ity property of Jacobi polynomials P(α,α) n (with respect to the weight function/parenleftbig 1−x2/parenrightbigα) on [−1,1], and using the Fourier-Hankel transform, one ﬁnds the relationships between anand ˆG, or between Aand ˆg, respectively, to be an= Λ n/integraldisplayx0 −1/parenleftbig 1−y2/parenrightbigαˆG(y)P(α,α) n (y)dy+ Λn/integraldisplay1 x0/parenleftbig 1−y2/parenrightbigαG(y)P(α,α) n (y)dy(2. 199) and A(λ) =λ1−2η/braceleftbigg/integraldisplay∞ 1rˆg(r)Jα(λr)dr+/integraldisplay1 0rg(r)Jα(λr)dr/bracerightbigg ,(2. 200) where Λn= 2−2α−1(2n+ 2α+ 1)Γ (n+α+ 1−η) Γ (n+α+ 1 +η)Γ (n+ 2α+ 1) Γ (n+ 1) Γ2(n+α+ 1). Substitute these expressions into (2 .192) and (2 .195), respectively, to obtain ﬁrst-kind Fredholm equations for the unknown functions ˆGand ˆg: /integraldisplayx0 −1ˆG(y)/parenleftbig 1−y2/parenrightbigαK(η) 1(x,y)dy=ˆF(x), x∈(−1,x0), (2. 201) /integraldisplay∞ 1ˆg(r)rK(η) 2(ρ,r)dr=ˆf(ρ), ρ∈(1,∞) (2. 202) where the functions ˆFand ˆfare explicitly calculated from ˆF(x) =F(x)−/integraldisplay1 x0G(y)/parenleftbig 1−y2/parenrightbigαK(η) 1(x,y)dy, (2. 203) ˆf(ρ) =f(ρ)−/integraldisplay1 0g(r)rK(η) 2(ρ,r)dr, (2. 204) and the kernels of these integral equations are K(η) 1(x,y) =∞/summationdisplay n=0ΛnP(α,α) n (x)P(α,α) n (y) (2. 205) and K(η) 2(ρ,r) =/integraldisplay∞ 0λ1−2ηJα(λρ)Jα(λr)dλ. (2. 206) ©200 1 CRC Press LLC Now we transform these kernels using the Abel integral representations for Jacobi polynomials P(α,α) n (1. 171) and Bessel functions Jα(1. 180): P(α,α) n (y) =(1 +y)−αΓ (n+α+ 1) Γ (η) Γ (n+α+ 1−η)/integraldisplayy −1(1 +u)α−ηP(α+η,α−η) n (u) (y−u)1−ηdu, (2. 207) Jα(λr) =ληrα 2η−1Γ (η)/integraldisplay∞ rv−α−η+1Jα+η(λv) (v2−r2)1−ηdv. (2. 208) We transform the kernel K(η) 1by substituting (2 .207) into (2 .205) and in- verting the order of summation and integration to ﬁnd K(η) 1(x,y) =2−2α−1 Γ (η)(1 +y)−α/integraldisplayy −1du(1 +u)α−η (y−u)1−ηk(η) 1(x,u), (2. 209) where k(η) 1(x,u) = ∞/summationdisplay n=0(2n+ 2α+ 1) Γ (n+ 1) Γ (n+ 2α+ 1) Γ (n+α+ 1) Γ (n+α+ 1 +η)P(α,α) n (x)P(α,α) n (u).(2. 210) The sum of the series in (2 .210) is a discontinuous function; when −1≤u<x, its value is [55] k(η) 1(x,u) = 22α+1{Γ (η)}−1(x−u)η−1(1−u)−α−η(1 +x)−α,(2. 211) and whenx<u ≤1,its value is zero. It follows that K(η) 1has the represen- tation K(η) 1(x,y) =(1 +x)−α(1 +y)−α Γ2(η)/integraldisplaymin(x,y) −1du(1−u)−α−η(1 +u)α−η (x−u)1−η(y−u)1−η. (2. 212) We transform the kernel K(η) 2by substituting (2 .208) into (2 .206) and in- terchanging the order of integration. The result is K(η) 2(ρ,r) =rα 2η−1Γ (η)/integraldisplay∞ rdvv−α−η+1 (v2−r2)1−η/integraldisplay∞ 0λ1−ηJα(λρ)Jα+η(λv)dλ. (2. 213) The inner integral in (2 .213) is the discontinuous Weber-Schafheitlin integral [19], [55]; when 0 ≤ρ<v, its value is /integraldisplay∞ 0λ1−ηJα(λρ)Jα+η(λv)dλ=ρα21−η{Γ (η)}−1/parenleftbig v2−ρ2/parenrightbigη−1v−α−η, (2. 214) ©200 1 CRC Press LLC and whenρ>v, its value is zero. Thus, the kernel K(η) 2can be expressed as K(η) 2(ρ,r) =ραrα 22η−2Γ2(η)/integraldisplay∞ max( ρ,r)v−2α−2η+1 (v2−ρ2)1−η(v2−r2)1−ηdv. (2. 215) The relationship between K(η) 1andK(η) 2can now be stated. Using the substitutions v= (1−u)1 2(1 +u)−1 2,ρ= (1−x)1 2(1 +x)−1 2,r= (1−y)1 2(1 +y)−1 2, we obtain K(η) 2/parenleftBig (1−x)1 2(1 +x)−1 2,(1−y)1 2(1 +y)−1 2/parenrightBig = (1−x)α 2(1 +x)α 2+1−η(1−y)α 2(1 +y)α 2+1−ηK(η) 1(x,y).(2. 216) The relationship between the pairs of Equations (2 .192) and (2 .193) and (2.194) and (2 .195) and their solutions anandAis now easily obtained, and the details are left to the reader. Before concluding this section, we draw the reader’s attention to one re- markable consequence of the kernel representations (2 .212) and (2 .215): we can ﬁnd the analytic solution to both integral Equations (2 .201) and (2 .202). If we substitute the kernel representation (2 .212) into (2 .201), it takes the form /integraldisplayx0 −1dy(1−y)αˆG(y)/integraldisplaymin(x,y) −1(1−u)−α−η(1 +u)α−η (x−u)1−η(y−u)1−ηdu = Γ2(η) (1 +x)αˆF(x), x∈(−1,x0).(2. 217) We split the interval of integration for the external integral; symbolically, this operation may be represented as /integraldisplayx0 −1=/integraldisplayx −1+/integraldisplayx0 x. (2. 218) Considering the ﬁrst integral on the right-hand side of (2 .218),the upper limit of the inner integral in (2 .217) is min ( x,y) =y(<x); for the second integral on the right-hand side of (2 .218),the upper limit of the inner integral in (2.217) is min ( x,y) =x(<y).Thus, the integral Equation (2 .217) becomes /integraldisplayx −1dy(1−y)αˆG(y)/integraldisplayy −1(1−u)−α−η(1 +u)α−η (x−u)1−η(y−u)1−ηdu+ /integraldisplayx0 xdy(1−y)αˆG(y)/integraldisplayx −1(1−u)−α−η(1 +u)α−η (x−u)1−η(y−u)1−ηdu = Γ2(η) (1 +x)αˆF(x), x ∈(−1,x0).(2. 219) ©200 1 CRC Press LLC Transform the ﬁrst term of the left-hand side of this equation using Dirich- let’s extended Formula (1 .135); invert the order of integration in the second term. These operations lead to /integraldisplayx −1du(1−u)−α−η(1 +u)α−η (x−u)1−η/integraldisplayx0 udy(1−y)αˆG(y) (y−u)1−η = Γ2(η) (1 +x)αˆF(x), x∈(−1,x0).(2. 220) Equation (2 .220) may be recognised as Abel’s integral equation /integraldisplayx −1G1(u)du (x−u)1−η= Γ2(η) (1 +x)αˆF(x), x∈(−1,x0), (2. 221) where the (as yet unknown) function G1is given by G1(u) = (1 −u)−α−η(1 +u)α−η/integraldisplayx0 udy(1−y)αˆG(y) (y−u)1−η. (2. 222) From the inverse Formula (1 .131), we deduce G1(u) = Γ2(η)sin (ηπ) πd du/integraldisplayu −1(1 +x)αˆF(x) (x−u)ηdx. (2. 223) Recognising that (2 .222) is also an Abel integral equation, the inversion Formula (1.133) leads to the ﬁnal and explicit form of the analytic solution to (2.201): ˆG(y) =−sin2(ηπ) π2Γ2(η) (1−y)−α× d dy/integraldisplayx0 ydu(1−u)α+η(1 +u)α−η (x−y)ηd du/integraldisplayu −1dx(1 +x)αˆF(x) (u−x)η.(2. 224) The solution of Equation (2. 202) can be obtained in a similar way, and the reader may wish to verify that ˆg(r) =−22η−2Γ2(η)sin2(ηπ) π2r−α−1× d dr/integraldisplayr 1dvv2α+2η (r2−v2)ηd dv/integraldisplay∞ vdρρ−α+1ˆf(ρ) (ρ2−v2)η.(2. 225) 2.6 Dual integral equations involving Bessel functions In this section we demonstrate how to apply Abel’s integral transform to obtain the solution of dual integral equations whose kernels are Bessel func- tions of ﬁxed order α. We shall treat two kinds of dual integral equations, the ©200 1 CRC Press LLC pair ∞/integraldisplay 0λ2ηA(λ)Jα(λρ)dλ=g(ρ),0≤ρ<1, (2.226) ∞/integraldisplay 0A(λ)Jα(λρ)dλ=f(ρ),ρ> 1, (2.227) andthecomplementarypair ,inwhichth esubintervalsofdeﬁnitionh avebeen interchanged, ∞/integraldisplay 0A(λ)Jα(λρ)dλ=f(ρ),0≤ρ<1, (2.228) ∞/integraldisplay 0λ2ηA(λ)Jα(λρ)dλ=g(ρ),ρ> 1, (2.229) whereAistheunkn ownfunctiontobedetermined.Theparameter ηsatisﬁes 0<η≤1 2,andg,faregi venfunctions,whichposses sFourier-Besseli ntegral expansions g(ρ)=∞/integraldisplay 0λ2ηG(λ)Jα(λρ)dλ, (2.230) f(ρ)=∞/integraldisplay 0F(λ)Jα(λρ)dλ. (2.231) DenotebyL2(µ)thespac eoffunctions Bdeﬁnedon[0 ,∞)satisfying ∞/integraldisplay 0λµ|B(λ)|2dλ<∞. Weshallﬁndth esolutionAofthes edualintegralequation sinthefunctional classL2(2η−1),assumin gthatthefunction sF,Gbelon gtothesameclass aswell: A,F,G ∈L2(2η−1). Aswehavepreviouslyremarked,th econditionimposedonthesolutionclass isareﬂectionoftheboundednessofth eenergycondition(Section1.3). ©200 1 CRC Press LLC Using the Formula (1 .181), we integrate Equation (2 .226) and obtain the dual equations ∞/integraldisplay 0λ−1+2η{A(λ)−G(λ)}Jα+1(λρ)dλ= 0,0≤ρ<1 (2. 232) ∞/integraldisplay 0{A(λ)−F(λ)}Jα(λρ)dλ= 0, ρ> 1. (2. 233) Now substitute for the Bessel functions occurring in these equations, the Abel integral representations derived from (1 .178) and (1 .180), Jα+1(λρ) =λ1−ηρ−α−1 2−ηΓ (1−η)ρ/integraldisplay 0vα+η+1Jα+η(λv) (ρ2−v2)ηdv, (2. 234) Jα(λρ) =ληρα 2η−1Γ (η)∞/integraldisplay ρv−α−η+1Jα+η(λv) (v2−ρ2)1−ηdv. (2. 235) Interchanging the order of integration, one obtains the following pair of ho- mogeneous Abel integral equations: ρ/integraldisplay 0vα+η+1 (ρ2−v2)η  ∞/integraldisplay 0λη{A(λ)−G(λ)}Jα+η(λv)dλ  dv= 0,0≤ρ<1 (2. 236) ∞/integraldisplay ρv−α−η+1 (v2−ρ2)1−η  ∞/integraldisplay 0λη{A(λ)−F(λ)}Jα+η(λv)dλ  dv= 0, ρ> 1. (2. 237) These equations possess unique zero solutions; the expressions in brackets therefore vanish, and we deduce a piecewise continuous representation of the sort that has repeatedly appeared in this book: ∞/integraldisplay 0ληA(λ)Jα+η(λρ)dλ=  ∞/integraltext 0ληG(λ)Jα+η(λρ)dλ, 0≤ρ<1 ∞/integraltext 0ληF(λ)Jα+η(λρ)dλ, ρ> 1.  (2. 238) Let us use the Hankel transform to reach the ﬁnal form of solution of these equations. Multiply both sides of (2 .238) by the factor ρJ α+η(µρ) and inte- ©200 1 CRC Press LLC grate over (0 ,∞) to obtain the closed form solution A(µ) =µ1−η1/integraldisplay 0dρ.ρJ α+η(µρ)∞/integraldisplay 0ληG(λ)Jα+η(λρ)dλ+ µ1−η∞/integraldisplay 1dρ.ρJ α+η(µρ)∞/integraldisplay 0ληF(λ)Jα+η(λρ)dλ. (2. 239) Notice that this solution is valid provided α>−1 2. The dual Equations (2 .228) and (2 .229) are solved in a similar way; the solution is A(µ) =µ1−η1/integraldisplay 0dρ.ρJ α−η(µρ)∞/integraldisplay 0ληF(λ)Jα−η(λρ)dλ+ µ1−η∞/integraldisplay 1dρ.ρJ α−η(µρ)∞/integraldisplay 0ληG(λ)Jα−η(λρ)dλ. (2. 240) Thus, both pairs of dual integral equations possess a closed-form analytical solution. More complicated dual integral equations may be transformed to second- kind Fredholm integral equations, provided some suitable and asymptoticallysmall parameters can be identiﬁed. For example, we may treat the dualequations ∞/integraldisplay 0λ2ηA(λ){1 +h(λ)}Jα(λρ)dλ=g(ρ),0≤ρ<1 (2. 241) ∞/integraldisplay 0A(λ){1 +p(λ)}Jα(λρ)dλ=f(ρ), ρ> 1 (2. 242) where the functions h,psatisfy lim λ→∞h(λ) = lim λ→∞p(λ) = 0. These conditions ensure that the integral operator in the equation is compact(completely continuous) in L 2(2η−1)(seeAppendixC.2).Inaddition ,the expansions (2 .230),(2.231) forf,gmust hold. Following the same steps used to obtain solution of (2 .226) and (2 .227), we obtain the second-kind Fredholm ©200 1 CRC Press LLC integralequation {1+p(µ)}A(µ)+µ1−η∞/integraldisplay 0ληA(λ){h(λ)−p(λ)}Kα+η(λ,µ)dλ =F(µ)+µ1−η∞/integraldisplay 0λη{G(λ)−F(λ)}Kα+η(λ,µ)dλ(2.243) wherethekerne lis Kα+η(λ,µ)=1/integraldisplay 0ρJα+η(λρ)Jα+η(µρ)dρ. (2.244) Thissecon dkin dequationenj oysthesameadvantageside ntiﬁedforthesecond kindmatrixsystemsobtaine dfordualserie sequations. 2.7Nonsymmetricaltripl eseriesequations InSection2.4wedescribe daneﬀectivealgorith mforth esolutionofsym- metrictripleserie sequations .Howmayon esolvesuchequation sinthemore generalcas ewhenth esubdivisionofthecomplet einter val[−1,1]ofdeﬁni- tionisnotsymmetric ?Theanswe rhasitsbasi sinresultsthatwerederived inSection2.5.Moreover,thesolutionoftripl eintegralequations ,involving Bessel functions, can be derived from the same results. First, we consider some particular (but frequently occurring in practice) equations involving associated Legendre functions Pm n(cosθ) or Bessel func- tionsJm(vρ).Subsequently, we will extend the method to equations involv- ing Jacobi polynomials P(α,α) n or Bessel functions Jα(λx) of arbitrary order as well. Letmbe a ﬁxed non-negative integer, and α,βbe ﬁxed so that 0 < α < β <π. Consider the following two sets of triple series equations.   ∞/summationtext n=mAm n(2n+ 1)Pm n(cosθ) =Fm 1(θ), θ∈(0,α) ∞/summationtext n=mAmnPm n(cosθ) =Fm 2(θ), θ∈(α,β) ∞/summationtext n=mAmn(2n+ 1)Pm n(cosθ) =Fm 3(θ), θ∈(β,π)(2. 245) ©200 1 CRC Press LLC and   ∞/summationtext n=mCm nPm n(cosθ) =Gm 1(θ), θ∈(0,α) ∞/summationtext n=mCm n(2n+ 1)Pm n(cosθ) =Gm2(θ), θ∈(α,β) ∞/summationtext n=mCm nPm n(cosθ) =Gm3(θ), θ∈(β,π)(2. 246) The solution {Am n,Cm n}∞ n=mof these triple series equations is sought in the functional class l2(2m).We consider only the ﬁrst set (2 .245), because the analysis of the equations (2 .246) is similar . On the basis of the relations (2 .175) and (2 .176) one may show that the triple Equations (2 .245) are equivalent to the following triple integral equa- tions, with Bessel function kernels,   ∞/integraltext 0λ.Am(λ)Jm(λρ)dλ=/parenleftbig 1 +ρ2/parenrightbig−3 2Fm 1(2 arctanρ),0≤ρ<ρ 0 ∞/integraltext 0Am(λ)Jm(λρ)dλ=/parenleftbig 1 +ρ2/parenrightbig−1 2Fm 2(2 arctanρ), ρ 0<ρ<ρ 1 ∞/integraltext 0λ.Am(λ)Jm(λρ)dλ=/parenleftbig 1 +ρ2/parenrightbig−3 2Fm 3(2 arctanρ), ρ 1<ρ (2. 247) whereρ0= tan1 2α, ρ 1= tan1 2β,andρ= tan1 2θ.The relation between the coeﬃcients Am nand the function Amis given by (2 .177),(2.178).The trans- formρ= tan1 2θmay be geometrically visualised as a stereographic projection of the sphere onto a plane. The “symmetrisation” of equations (2 .247), pro- ducing a symmetric partition of the domain of deﬁnition for each functional equation of the set, is realised by an “inversion in a circle.” Introduce thenew variable r= (ρ 0ρ1)−1 2ρ= (ρ0ρ1)−1 2tan1 2θ, (2. 248) so that θ=θ(r) = 2 arctan/bracketleftBig (ρ0ρ1)1 2r/bracketrightBig , and transform Equations (2 .247) to   ∞/integraltext 0µ.A 1(µ)Jm(µr)dµ=ρ0ρ1(1 +ρ0ρ1r2)−3 2Fm 1{θ(r)},0≤r<r 0 ∞/integraltext 0A1(µ)Jm(µr)dµ= (ρ0ρ1)1 2/parenleftbig 1 +ρ0ρ1r2/parenrightbig−1 2Fm 2{θ(r)}, r 0<r<r 1 ∞/integraltext 0µ.A 1(µ)Jm(µr)dµ=ρ0ρ1/parenleftbig 1 +ρ0ρ1r2/parenrightbig−3 2Fm 3{θ(r)}, r 1<r (2. 249) whereµ= (ρ0ρ1)1 2λ, r 0= (ρ0/ρ1)1 2=r−1 1,andA1(µ) =A(λ). ©200 1 CRC Press LLC After the ﬁnal change of variables r= tan1 2ϑ= (ρ0ρ1)−1 2tan1 2θ, (2. 250) so that θ=θ(ϑ) = 2 arctan/bracketleftbigg (ρ0ρ1)1 2tan1 2ϑ/bracketrightbigg , (which may be visualised geometrically as reconstruction of the spherical sur- face from its stereographic projection in the plane), we obtain the following symmetric triple series equations, involving the associated Legendre functions Pm nas kernels:   ∞/summationtext n=m(2n+ 1)Bm nPm n(cosϑ) =ρ0ρ1u(ϑ)−3 2Fm 1(θ(ϑ)), ϑ ∈(0,ϑ0) ∞/summationtext n=mBm nPm n(cosϑ) = (ρ0ρ1)1 2u(ϑ)−1 2Fm 2(θ(ϑ)), ϑ ∈(ϑ0,π−ϑ0) ∞/summationtext n=m(2n+ 1)Bm nPm n(cosϑ) =ρ0ρ1u(ϑ)−3 2Fm 3(θ(ϑ)), ϑ∈(π−ϑ0,π) (2. 251) whereu(ϑ) = (cos21 2ϑ+ρ0ρ1sin21 2ϑ),ϑ0= 2 arctan ( ρ0/ρ1)1 2,and so cosϑ0= (ρ1−ρ0)/(ρ1+ρ0).Note that in deriving (2 .251) we used a relationship comparable to (2 .176): ∞/summationdisplay s=m(2s+ 1)Bm sPm s(cosϑ) = sec3ϑ 2∞/integraldisplay 0µA1(µ)Jm/parenleftbigg µtanϑ 2/parenrightbigg dµ (2. 252) Using (2.252) and (2 .175),the relationship between the coeﬃcients Am nand Bm nis Am n=1√ 2(n−m)! (n+m)!1/integraldisplay −1dxPm n{z(x)}/radicalbig 1 +ρ0ρ1+ (1−ρ0ρ1)x∞/summationdisplay s=m(2s+ 1)Bm sPm s(x) (2. 253) where z(x) =1−ρ0ρ1+ (1 +ρ0ρ1)x 1 +ρ0ρ1+ (1−ρ0ρ1)x. It is obvious that if ρ1=ρ−1 0, thenAm n≡Bm n,and the Equations (2 .251) will be identical with (2 .245). The relation between Jacobi polynomials P(m,m ) n−mand associated Legendre functionsPm n, Pm n(z) = 2−m/parenleftbig 1−z2/parenrightbigm 2Γ (n+m+ 1) Γ (n+ 1)P(m,m ) n−m(z) ©200 1 CRC Press LLC enablesustoconvert(2 .251)toth etypeofsymmetrictripleseriesequations solvedinSection2.4.Thus ,thelinear-fractionaltransform cosθ=z=1−ρ0ρ1+ (1 +ρ0ρ1)x 1 +ρ0ρ1+ (1−ρ0ρ1)x, (2. 254) withx= cosϑ, “symmetrises” the initial Equations (2 .245) and converts them to the symmetric triple series Equations (2 .251) with a new set of unknown coeﬃcients {Bs}∞ s=m, for which the relationship with the original set of un- knowns {Am n}∞ n=mis given by the Formula (2 .253). From this derivation, one further result should be noted: the triple integral equations involving the Bessel functions Jm(λρ),with arbitrary fragmenta- tion of the complete range of the variable ρ,may be transformed to a set of triple series equations involving Pm n(cosθ), with symmetrical fragmentation of the corresponding interval. Triple series equations involving the trigonometric functions (sin nϕor cosnϕ) can be solved in an analogous manner. Although other methods have previ- ously been reported in the literature, the attractive approach suggested here is based on [77], [3], and [37]. Consider the triple series equations   ∞/summationtext n=1nansinnϕ=f1(ϕ), ϕ ∈(0,ϕ0) ∞/summationtext n=1ansinnϕ=f2(ϕ), ϕ ∈(ϕ0,ϕ1) ∞/summationtext n=1nansinnϕ=f3(ϕ), ϕ ∈(ϕ1,π)(2. 255) wheref1,f2,andf3are given functions, and φ0,φ1are ﬁxed so that 0 <φ 0< φ1<π. From (2.189) and (2 .190) it can be shown that Equations (2 .255) are equivalent to the following triple integral equations:   ∞/integraltext 0µA(µ) sin (µx)dµ=/parenleftbig 1 +x2/parenrightbig−1f1(2 arctanx), x<x 0 ∞/integraltext 0A(µ) sin (µx)dµ=f2(2 arctanx), x 0<x<x 1 ∞/integraltext 0µA(µ) sin (µx)dµ=/parenleftbig 1 +x2/parenrightbig−1f3(2 arctanx), x 1<x(2. 256) wherex= tan1 2ϕ,x 0= tan1 2ϕ0,andx1= tan1 2ϕ1.The unknown coeﬃcients {an}∞ n=1and function Aare related by an=8 πn∞/integraldisplay 0 ∞/integraldisplay 0µA(µ) sin (µu)dµu 1 +u2Uu−1/parenleftbigg1−u2 1 +u2/parenrightbigg du (2. 257) ©200 1 CRC Press LLC and A(µ) =1 π1/integraldisplay −1/braceleftBigg∞/summationdisplay n=1nanUn−1(z)/bracerightBigg/radicalbigg 1−z 1 +zsin/parenleftBigg µ/radicalbigg 1−z 1 +z/parenrightBigg dz (2. 258) whereUn(cosϕ) = sin (n+ 1)ϕ/sinϕis the Chebyshev polynomial of the second kind. Applying the change of variables connected with inversion in a circle y= (x0x1)−1 2x, v = (x0x1)1 2µ, so that ϕ=ϕ(y) = 2 arctan/bracketleftBig (x0x1)1 2y/bracketrightBig , the triple Equations (2 .256) become the symmetric triple integral equations   ∞/integraltext 0vA1(v) sin(vy)dv=x0x1/parenleftbig 1 +x0x1y2/parenrightbig−1f1{ϕ(y)}, y<y 0 ∞/integraltext 0A1(v) sin(vy)dv= (x0x1)1 2f2{ϕ(y)}, y 0<y<y 1 ∞/integraltext 0vA1(v) sin(vy)dv=x0x1/parenleftbig 1 +x0x1y2/parenrightbig−1f3{ϕ(y)}, y 1<y (2. 259) wherey0=/parenleftbig tan1 2ϕ0cot1 2ϕ1/parenrightbig1 2,y1=y−1 0andA1(v) =A(µ). Using the transform ϑ= 2 arctany,so that ϕ=ϕ(ϑ) = 2 arctan/bracketleftbigg (x0x1)1 2tan1 2ϑ/bracketrightbigg , and the relationships ∞/summationdisplay n=1nbnsinnϑ= sec2ϑ 2∞/integraldisplay 0vA1(v) sin (vy)dv, (2. 260) ∞/summationdisplay n=1bnsinnϑ=∞/integraldisplay 0A1(v) sin (vy)dv, (2. 261) one may reduce Equations (2 .259) to the following symmetric series equations with trigonometric kernels, to be solved for the new set of unknowns {bn}∞ n=1:   ∞/summationtext n=1nbnsinnϑ=x0x1/parenleftbig cos21 2ϑ+x0x1sin21 2ϑ/parenrightbig−1f1{ϕ(ϑ)}, ϑ∈(0,ϑ0) ∞/summationtext n=1bnsinnϑ= (x0x1)1 2f2{ϕ(ϑ)}, ϑ ∈(ϑ0,π−ϑ0) ∞/summationtext n=1nbnsinnϑ=x0x1/parenleftbig cos21 2ϑ+x0x1sin21 2ϑ/parenrightbig−1f3{ϕ(ϑ)}, ϑ∈(ϑ0,π)(2. 262) ©200 1 CRC Press LLC whereϑ0=2arctan/parenleftbig tan1 2ϕ0cot1 2ϕ1/parenrightbig .From(2.260)an d(2.257)weobtain therelationshipbetwee nthetwoset sofcoeﬃcie nts: an=4 πn1/integraldisplay −1dz√ 1−z2 1+x0x1+(1−x0x1)z× Un−1/braceleftbigg1−x0x1+(1+x0x1)z 1+x0x1+(1−x0x1)z/bracerightbigg∞/summationdisplay s=1sbsUs−1(z).(2.263) Duetothesymmetricalsubdivisionofthecomplet einter val[0,π],thesys- temofEquations(2 .262)maybesolvedbythemethoddevelope dinSection 2.4,byreducingittotwodecoupleddualseriesequations. 2.8Coupledserie sequations Coupledsystemsaris einse veralcontextsincludin gelasticity.Arecent exampleisthecrackanalysisofMarti n[39] .Althoughcoupledsystemswill bebrieﬂyencountere dinSection7.5,somegeneralconsiderationofthemis includedforcompleteness.Thus ,weconsidercouple dseriesequation softhe followingtype, ∞/summationdisplay n=0λn(α,β;η){a(1−rn)xn+b(1−sn)yn}P(α,β) n(x)=F1(x), (2.264) ∞/summationdisplay n=0λn(γ,δ;ε){c(1−tn)xn+d(1−un)yn}P(γ,δ) n(x)=F2(x), (2.265) ∞/summationdisplay n=0(1−pn)xnP(α,β) n(x)=G1(t), (2.266) ∞/summationdisplay n=0(1−qn)ynP(γ,δ) n(x)=G2(x), (2.267) whereth eﬁrstpai rholdsforx∈(−1,x0),andth esecon dpairhold sforx∈ (x0,1).Theunknowns xn,ynaretobefound ;theparameter sα,β,η,γ,δ,ε obeytheconstraintcondition sofSection2.1an dη,ε∈(0,1).Thesequence termsrn,sn,tn,un,pn,qnvanish asn→ ∞.The right-hand sides of these ©200 1 CRC Press LLC equationsh aveFourier-Jacobiexpansions F1(x)=∞/summationdisplay n=0λn(α,β;η)f1 nP(α,β) n(x), (2.268) F2(x)=∞/summationdisplay n=0λn(γ,δ;ε)f2 nP(γ,δ) n(x), (2.269) G1(x)=∞/summationdisplay n=0g1 nP(α,β) n(x), (2.270) G2(x)=∞/summationdisplay n=0g2 nP(γ,δ) n(x). (2.271) Thefollowingregularisationprocedureisjustiﬁe dbyth esam esortofargu- mentsasemployedinSection2.1,an dsoweomi tanydiscussionofthi saspect, and present the formal technique. The systems are nontrivially coupled pro- videdbc/negationslash= 0.Without loss of generality, we may suppose that pn=qn= 0. Multiply (2. 264) by (1 + x)βand integrate, then use the integral representa- tion of Abel type (1. 172) to obtain ∞/summationdisplay n=0cn/braceleftbig a(1−rn)xn+b(1−sn)yn−f1 n/bracerightbig P(α−η,β+η) n (x) = 0, x∈(−1,x0),(2. 272) where cn=Γ (α+n+ 1) Γ (α+n+ 1−η); similarly, multiply (2. 264) by (1 + x)δand integrate, then use the integral representation (1. 172) to obtain ∞/summationdisplay n=0dn/braceleftbig c(1−tn)xn+d(1−un)yn−f2 n/bracerightbig P(γ−ε,δ+ε) n (x) = 0, x∈(−1,x0),(2. 273) where dn=Γ (γ+n+ 1) Γ (γ+n+ 1−ε). On the other hand, using the integral representation (1. 171) for P(α,β) n andP(γ,δ) n,we obtain ∞/summationdisplay n=0/parenleftbig xn−g1 n/parenrightbig cnP(α−η,β+η) n (x) = 0, x∈(x0,1), (2. 274) ©200 1 CRC Press LLC ∞/summationdisplay n=0/parenleftbig yn−g2 n/parenrightbig dnP(γ−ε,δ+ε) n (x) = 0, x∈(x0,1). (2. 275) Rearrange (2. 272) and (2. 274) in the form ∞/summationdisplay n=0axncnP(α−η,β+η) n (x) =/braceleftbigg H1(x), x∈(−1,x0) H2(x), x ∈(x0,1)/bracerightbigg , (2. 276) where H1(x) =∞/summationdisplay n=0cn/braceleftbig arnxn−b(1−sn)yn+af1 n/bracerightbig P(α−η,β+η) n (x), (2. 277) H2(x) =∞/summationdisplay n=0cnag1 nP(α−η,β+η) n (x); (2. 278) similarly, rearrange (2. 273) and (2. 275) in the form ∞/summationdisplay n=0dndynP(γ−ε,δ+ε) n (x) =/braceleftbiggH3(x), x∈(−1,x0) H4(x), x ∈(x0,1)/bracerightbigg (2. 279) where H3(x) =∞/summationdisplay n=0dn/braceleftbig dunyn−c(1−tn)xn+f2 n/bracerightbig P(γ−ε,δ+ε) n (x), (2. 280) H4(x) =∞/summationdisplay n=0dn/braceleftbig dg2 n/bracerightbig P(γ−ε,δ+ε) n (x). (2. 281) A standard orthogonality argument produces the coupled i.s.l.a.e. adiag(1 −rn)Ix+aK1x+bK2y=f, ddiag(1 −un)Ix+cK3x+dK4y=g, wherex={xn}∞ n=1,y={yn}∞n=1,diag(1 −rn) and diag(1 −un) denote di- agonal operators formed from the sequences {rn}∞ n=1,{un}∞n=1,Idenotes the identity operator, and K1,K2,K3,K4denote compact operators whose ma- trix entries are calculated in terms of unnormalised incomplete scalar products Q(γ−ε,δ+ε) nm (x0) andQ(α−η,β+η) nm (x0); alsof,gare explicitly known. Provided ad/negationslash= 0,the system is a Fredholm system of second kind; numerically, when the truncation method is used, it has the same advantages as previously noted for uncoupled systems. ©200 1 CRC Press LLC 2.9Aclasso fintegro-seriesequations Theapproachdevelopedintheprevioussectionsprovidesauniﬁedtreat- mentforbothseriesandintegralequations .It,therefore ,provide sperhaps themostsuitablefoundationforinvestigatingacertai nclassoffunctional equations,theso-called integro-series equations(I.S.E.) .Thisnovelclass arisesfrommixedboundaryvalueproblemsinpotentialtheoryo rdiﬀraction forstructurescomposedofplan eorcurvilinearconductingsurfaces.Letus brieﬂydescribethetypeofequationsi nthisclass,bu tdeferfurthe rdescrip- tionofsolutiontechniquesu ntilSection8.5,whereaspeciﬁcproble mofthis type concerning a spherical cap and a circular disc, will be encountered. In operator notation, the integro-series equations take the form L11(u){A(µ)}+L12{v(u)}{Bn}=F1(u), a≤u≤c,(2. 282) L22(u){Bn}+L21{u(v)}{A(µ)}=F2(v), α≤v≤γ,(2. 283) where Li1{A(µ)}=c/integraldisplay aA(µ)/braceleftBigg K(1) i1(u,µ), a≤u<b K(2) i1(u,µ), b<u ≤c/bracerightBigg dµ, i = 1,2 (2. 284) and Li2(v){Bn}=∞/summationdisplay n=0Bn/braceleftBigg K(1) i2(v,n), α≤v<β K(2) i2(v,n), β <v ≤γ/bracerightBigg , i = 1,2.(2. 285) The solution of the I.S.E. is sought in the standard functional space: {Bn}∞ n=0∈l2, andA∈L2(a,c) . (2. 286) The main technical diﬃculty encountered in solving these equations is the expansion of the kernels deﬁned in (2 .284) and (2 .285), in terms of eigen- functions of the Laplace operator in some other coordinate system. Using the relations connecting diﬀerent coordinate systems, in which the considered shells are described intrinsically as parts of coordinate surfaces u=u(v), v=v(u),these re-expansions take the form K(1,2) i1(u(v),µ) =∞/summationdisplay n=0C(1,2) n(µ)K(1,2) i2(v,n) (2. 287) and K(1,2) i2(v(u),u) =c/integraldisplay aD(1,2) u(µ)K(1,2) i1(u,µ)dµ (2. 288) ©200 1 CRC Press LLC The substitution of (2 .287) and (2 .288) into (2 .282)–(2.285), and application of Abel’s integral equation method leads ultimately to an I.S.E. of the second kind, which is a perturbation of the identity by a completely continuous oper-ator, in the Cartesian product of functional spaces l 2×L2(a,c).The method is valid for arbitrary location of shells that make no contact or intersection. When the (imaginary) continuation of that coordinate surface that describes the open shell intersects the real surface of another shell, some technical dif-ﬁculties may appear. These diﬃculties are not insurmountable and can beovercome by a correct representation of the desired solution. ©200 1 CRC Press LLC Chapter3 ElectrostaticPotentialTheor yfor OpenSphericalShells Sphericalgeometryprovidesthesimplestandmostattractivesettingforthree- dimensionalpotentialtheory.Theelectrostati cpotentialsurroundingaclosed conductingspher eonwhi chth esurfacepote ntialiss peciﬁe diseasil ycalcu- latedi ntermsofsphericalharmonics;ithasanes peciall ysimplefor mifthe surfaceisanequipote ntialsurface.Whenaperturesareintroduced ,someof thissimplicityi sretainedprovidedthesurfac eispuncturedinarotationally symmetricfashion. Asinglecircularaperture,characterisedbyth eangleθ1itsubtendsatthe centreofth esphericalstructure ,isth etopologicall ysimples tsuchstructure, thoughratherdiﬀerentformsoftheshellap pearasθ1varies,fromthenearly enclosedsphericalcavity( θ1→0)throughanopensphericalcap(0 <θ1<π) toaslightlycurvedcirculardis c(θ1→π).Closed-formsolutionsthatcanbe obtainedforthisfamil yofshellsbysolvinganappropriatese tofdualseries equations,arepresentedinSection3.1. Closed-formsolutionsdonotexis tformorecomplicatedshellstructures, suchastheaxisymmetricsphericalbarrel(i nwhichthesphericalsurfac eis puncturedbytwoequalcircularholes)orthecompleme ntarysurface,apair ofsphericalcaps.Perhap sthebestcriterionbywhichtojudgeasolutionisits accuracyandeﬀecti venessfornumericalcalculation.Th epotentialproblem forthebarre l(orcaps)m aybeformulatedastripl eserie sequations ;thereg- ularisationan dconversiont oasecond-kindFredhol mmatrixsyste mprovides anexcellentbasisforbothappr oximateanalyticalestimate saswellasprecise numericalcalculationbecaus ethenor mofth ecompac toperatoroccurringthe resultingsyste missmall(rathe rlessthanunity) .Thus,th eimpac tofedges andtheinﬂuenc eofth ecavityonthepote ntialdistributioncanbeassessed withrelativeease.Someexample softh epotentialdistributionaroundthese structuresaregiveninSectio n3.2,togetherwithcapacitanceestimatesfor thecondensorforme dfromanoppositel ycharge dpairofcaps.Section3.3 extendsth etripleseriesapproachtoabarrelwithunequalholes(bu tlocated axisymmetrically),andt oitscomplementarysurface ,apairofunequallysized sphericalcaps. Section3. 4consider spairsofsphericalcap swhi chlieondiﬀerentbu ttouch- ing spheres. The classical tool of inversion (in an appropriate sphere) producesplanar structures. The potential distribution may be described by dual in- ©200 1 CRC Press LLC Figur e3.1 Spherica lshellgeometry :(a)thespherica lcap,(b)asymmetrical pairofspherica lcaps,and(c)asymmetrica lspherica lbarrel. tegra lequations ;thesemayberegularise dtoproduceasyste mthatiswell suitedtoeﬀecti venumerica lcalculation. Avariantofthebarre lstructure salread yconsidere dprovidesamodelfor atypeofelectroni clens;thisisdiscusse dinsomedetai linSectio n3.5. Theﬁnaltwosection s(3.6and3.7)provideacontrasttotheprevious sections .Themagnetostati cpotentialsurroundin gsuperconductin gsurfaces givesrisetomixedboundar yvalueproblems ,butNeuman n(rathe rthan Dirichlet)boundar ycondition sareenforce donthespherica lsurface .How- ever,theresultin gseriesequation sareamenabl etothestandar dapproa ch developedinthischapte rforspherica lgeometr y,andthemagneti cﬁeldis determine dinsideaspherica lshell. 3.1Theopenconductin gspherica lshell Thespatia ldistributio noftheelectrostati cpotentialsurroundin gacharged spherica lcaphasbeeninvestigate dbymanyauthor s[41],[11],and[25]. Asmentione dintheintroduction ,itprovidesoneofthesimples tthree- dimensiona lmixedboundar yvalueproblem sforLaplace’ sequation .Inthis section ,wereformulatethiswell-kn ownproble mintermsofdualequation sin- volvin gJacob ipolynomials .Thetechnique sdescri bedinChapte r2providea standar dmeth odforthedeductio nofthesolution ;furthermore ,theyprovide arationa lbasisfromwhichmorecomplicate dproblem smaybetackled. LetU0=U0(θ,ϕ)betheelectrostati cpotentialthatisassume dtobe knownonthespherica lcap,ofradiu saandsubtendin ganangleθ0atthe spherica lcentre(seeFigur e3.1).Theonlyrequireme ntonthefunctio nU0is ©200 1 CRC Press LLC/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0 /1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0 /1/1/1/1 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/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0 /1/1/1/1/1/1/0/0/0 /1/1/1/0/0/0 /1/1/1 /0/0/0/0/1/1/1/1 /0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1z θ0z θ0z θ0a) b) c) o oo xy xy xy thati thasaFourier-Legendr eserie sexpansion: U0(θ,ϕ)=∞/summationdisplay m=0/parenleftbig 2−δ0 m/parenrightbig cosm(ϕ−ϕ0)∞/summationdisplay n=mαm nPm n(cosθ), (3.1) whereαm nareknown(Fourier )coeﬃcients. Weseekapote ntialU(r,θ,ϕ ),thatsatisﬁesth eLaplaceequation,iscontin- uousacros stheclosedsphericalsurface r=a,anddecay satinﬁnityaccording toU(r,θ,ϕ )=O/parenleftbig r−1/parenrightbig asr→∞.Thus,forsuitable Am n(tobedetermined), Uhastheform ∞/summationdisplay m=0/parenleftbig 2−δ0 m/parenrightbig cosm(ϕ−ϕ0)∞/summationdisplay n=mAm nPm n(cosθ)/braceleftbigg(r/a)n,0≤r<a (r/a)−n−1,r>a/bracerightbigg . (3.2) Itisclearthat,withn olossofgenerality,wemayassume ϕ0=0.Themixed boundaryconditionstobeenforce donthesphericalsurface r=aare(for ϕ∈(0,2π)) U(a,θ,ϕ )=U0(θ,ϕ),θ ∈(0,θ0), (3.3) /bracketleftbigg∂ ∂rU(r,θ,ϕ )/bracketrightbiggr=a+0 r=a−0=0,θ ∈(θ0,π). (3.4) Thelatte rcondition(3 .4)reﬂectsthecontinuityofth enormalderivativeof thepotentialfunctionacrosstheaperture .Du etoth ecompletenes sand orthogonalityofth esetoftrigonometricfunctions {cosmϕ}∞ m=0on(0,2π), thesolutionforeachindex mmaybeconsidere dinde pende ntly.Enforcing themixedboundarycondition sleadst othedualserie sequations ∞/summationdisplay n=mAm nPm n(cosθ)=∞/summationdisplay n=mαm nPm n(cosθ),θ∈(0,θ0),(3.5) ∞/summationdisplay n=m(2n+1)AmnPm n(cosθ)=0,θ ∈(θ0,π).(3.6) Letu sdetermineth esolutionclas sforthecoeﬃcients Am n,guidedbythe boundednessconditionforenergyi ntegral(Section1.3).Theintegrationre- gion is most conveniently chosen as the sphere of radius a, so that W=2π/integraldisplay 0dϕa/integraldisplay 0dr.r2π/integraldisplay 0dθsinθ|gradU|2<∞. (3. 7) It follows from (3 .7) that the solution class is deﬁned by ∞/summationdisplay m=0/parenleftbig 2−δ0 m/parenrightbig∞/summationdisplay n=mn 2n+ 1Γ (n+m+ 1) Γ(n−m+ 1)|Am n|2<∞. (3. 8) ©200 1 CRC Press LLC Usingthewell-kn ownrelationshi pbetweenPm nandultraspherica lpoly- nomial s(2.91)wemayreduc e(3.5)and(3.6)tothefollowingdualseries equations ,involvin gJacob ipolynomial saskernels(settin gn−m=s): ∞/summationdisplay s=0Xm s+m(s+m+1 2)P(m,m ) s (z)=0,z∈(−1,z0) (3.9) ∞/summationdisplay s=0Xm s+mP(m,m ) s (z)=∞/summationdisplay s=0βm s+mP(m,m ) s (z),z∈(z0,1) (3.10) wherez=cosθ,z0=cosθ0and /braceleftbiggXm s+m βm s+m/bracerightbigg =Γ(s+2m+1) Γ(s+m+1)/braceleftbiggAm s+m αm s+m/bracerightbigg . (3.11) Equation softhiskindarereadil ysolvedbythetechnique soutline din Chapte r2.Equation s(3.9)and(3.10)mayberecognise dasequation softhe form(2.4),(2.5)withtheidentiﬁcatio nα=β=m,fn=qn=rn=0,and gnreplace dbyβn+m;furthermore ,η=1 2,andλs/parenleftbig m,m;1 2/parenrightbig =s+m+1 2. From(2.23),wededuc ethattheanalytica lsolutio nis Ym p+m=∞/summationdisplay s=0ˆβm s+mˆQ(m−1 2,m+1 2) sp (z0) (3.12) wherep=0,1,2,...,and /braceleftBig Ym p+m,ˆβm p+m/bracerightBig =Γ(p+m+1) Γ(p+m+1 2)/bracketleftBig h(m−1 2,m+1 2) p/bracketrightBig1 2/braceleftbig Xm p+m,βm p+m/bracerightbig .(3.13) Thenormalise dincomplet escalarproductˆQ(α,β) sp(z0)isdeﬁne dby(2.24). Inconclusion ,itshoul dbenotedthatthesolutio nbelong stotherequired class(3.8);thiscanbeprovedusingtheproperties(B.171)and(B.172)of thefunctio nˆQ(α,β) sp(seeAppendixB.6). Whenthecapisanequipotentialsurface ,onlytheindex0coeﬃcie ntis nonzero ,andthesummatio n(3.12)comprise sasingleterm;thesolution simpliﬁe stothatalread yobtaine dinSectio n1.4. 3.2 A symmetrical pair of open spherical caps and the spherical barrel The most striking feature of the problem considered in the previous section is that an analytical, closed form of the electrostatic potential was obtained; ©200 1 CRC Press LLC thissolutionisobviouslyinde pende ntofth emethodused .Suchasimpl eand satisfactorysolutioncannotbeex pecte dformorecomplicatedconductors, suchasapairofchargedsphericalcaps,oracharge dsphericalshellwithtwo holes(theso-calledspherical barrel ).Thesestructuresprese ntveryparticular casesof two-body problemsinphysics ,inwhichthegoalistocalculateeﬀec- tivelythemutualimpac toftwobodies .Inageneralsituation ,themethod ofsuccessiveapproximationsisused.However ,thisi seﬀecti veinonlyafew situations,forexample,objectswithdimensionsver ymu chsmallerthantheir separation.Suchsituationsaresomewhatexceptional.Howe ver,ifth ebodies areidenticalthereisahighdegreeofsymmetryintheirmutualimpact,so thatthereissom ehopeofdescribin gthedominantpartofthei rinteraction analytically,e venwhenthe yareverycloselycoupled. Inthissectionweconsidertwoexample softhishighlysymmetricsituation; weproducesemi-analyti csolution sforth eelectrostaticpote ntialaroun da pairofsymmetricallyl ocated ,charge dsphericalcapsandaroundaspherical barrel.Theapproachiscompletel ybase donth eeﬀectiveprocedur eofsolving tripleseriesequationsi nvolvingLegendrepolynomials,describe dinSection 2.4. Thegeometr yisshowninFigures3.1ban d3.1c.Twosphericalcapsoccupy the region r=a, θ∈(0,θ0)∪(π−θ0,π), whilst the barrel occupies the complementary portion of the spherical surface deﬁned by r=a, θ∈(θ0,π−θ0). The conditions that the potential satisﬁes are similar to those for a singlesphericalcap(seeSection3.1) ,exceptthatthegivenpote ntialisn owas- sumed to be constant over each conductor surface. We seek the rotationallysymmetric potential Uin the form U=U(r,θ) = ∞/summationdisplay n=0xnPn(cosθ)/braceleftbigg(r/a)n,0≤r<a (r/a)−n−1,r>a/bracerightbigg (3. 14) where the unknown coeﬃcients {xn}∞ n=0satisfy (cf. (3. 8)) W= 4πa∞/summationdisplay n=1n 2n+ 1|xn|2<∞, (3. 15) so that {xn}∞n=0lies in the solution class l2≡l2(0). First let us consider the pair of charged caps, the upper (in the region z > 0) and lower being maintained at potentials 1 and ±1, respectively. Enforcement of the mixed boundary conditions leads to the symmetric triple ©200 1 CRC Press LLC series equations   ∞/summationtext n=0xnPn(z) = (−1)l, z ∈(−1,−z0) ∞/summationtext n=0/parenleftbig n+1 2/parenrightbig xnPn(z) = 0, z ∈(−z0,z0) ∞/summationtext n=0xnPn(z) = 1, z ∈(z0,1)(3. 16) wherez= cosθ, z 0= cosθ0,and the index ltakes the values 0 or 1 . Wemayusetheapproa chdescri bedinSectio n2.4,toreduc e(3.16)tothe following dual series equations involving the Jacobi polynomials P(0,l−1 2) n   ∞/summationtext n=0/parenleftbig n+1 2l+1 4/parenrightbig x2n+lP(0,l−1 2) n (u) = 0, u ∈(−1,u0), ∞/summationtext n=0x2n+lP(0,l−1 2) n (u) = (−1)l/braceleftbig1 2(1 +u)/bracerightbig−l 2, u ∈(u0,1),(3. 17) whereu= 2z2−1 andu0= 2z2−1 = cos 2θ0.These equations are now transformed in the standard way to the following inﬁnite systems of linear algebraic equations (i.s.l.a.e.) of the second kind. Denoting b2n+l=Γ (n+ 1) Γ/parenleftbig n+1 2/parenrightbig/braceleftbigg h(−1 2,l) n/bracerightbigg−1 2 x2n+l (3. 18) and εl n= 1−/parenleftbigg n+l 2+1 4/parenrightbiggΓ/parenleftbig n+l+1 2/parenrightbig Γ/parenleftbig n+1 2/parenrightbig Γ (n+l+ 1) Γ (n+ 1), (3. 19) so thatεl n=O/parenleftbig n−2/parenrightbig asn→ ∞ , the system for the even ( l= 0) coeﬃcients is /parenleftbig 1−ε0 m/parenrightbig b2m+∞/summationdisplay n=0b2nε0nˆQ(−1 2,0) nm (u0) =23 4√πˆQ(−1 2,0) 0m (u0) (3. 20) wherem= 0,1,2,...; the system for the odd ( l= 1) coeﬃcients is /parenleftbig 1−ε1 m/parenrightbig b2m+1+∞/summationdisplay n=0b2n+1ε1nˆQ(−1 2,1) nm (u0) =−2/braceleftBigg 1−u0 π/parenleftbig m+1 2/parenrightbig (m+ 1)/bracerightBigg1 2 ˆP(1 2,0) m (u0),(3. 21) wherem= 0,1,2,...; recall that the incomplete scalar products ˆQ(α,β) nm are deﬁned by Formula (2 .24). It is convenient to rearrange these second-kind systems by replacing the angle parameter u0(orθ0) byu1=−u0/parenleftbig orθ1=π 2−θ0/parenrightbig ,and using Equation ©200 1 CRC Press LLC (B.170)(seeAppendix )totransfor mtheincomplet escalarproducts .This leads to the following equivalent i.s.l.a.e., in which the index 0 equations have been separated out. Let c2m+l= (−1)mb2m+lwherel= 0 or 1. (3. 22) The even index system is /braceleftbigg 1−ε0 0ˆQ(0,−1 2) 00 (u1)/bracerightbigg c0= 23 4√π/braceleftbigg 1−ˆQ(0,−1 2) 00 (u1)/bracerightbigg +∞/summationdisplay n=1c2nε0 nˆQ(0,−1 2) n0 (u1),(3. 23) and c2m−∞/summationdisplay n=1c2nε0 nˆQ(0,−1 2) nm (u1) =c0ε0 0ˆQ(0,−1 2) 0m (u1)−23 4√πˆQ(0,−1 2) 0m (u1).(3. 24) form= 1,2,.... The odd index system comprises /braceleftbigg 1−ε1 0ˆQ(1,−1 2) 00 (u1)/bracerightbigg c1= −2√ 2√π(1 +u1)1 2ˆP(1,−1 2) 0 (u1) +∞/summationdisplay n=1c2n+1ε1 nˆQ(1,−1 2) n0 (u1),(3. 25) and c2m+1−∞/summationdisplay n=1c2n+1ε1 nˆQ(1,−1 2) nm (u1) = c1ε1 0ˆQ(1,−1 2) 0m (u1)−2/braceleftBigg 1 +u1 π/parenleftbig m+1 2/parenrightbig (m+ 1)/bracerightBigg1 2 ˆP(0,−1 2) m (u1).(3. 26) form= 1,2,.... Formulae (3 .23) and (3.25) provide the values of c0and c1for replacement in (3 .24) and (3.26) producing systems for {c2n}∞ n=1and {c2n+1}∞n=1.Bounds, which are uniform in the parameter u1,on the norms p andqof the completely continuous operators of the systems (3 .24) and (3.26) are q≤max/vextendsingle/vextendsingleε1 n/vextendsingle/vextendsingle=/vextendsingle/vextendsingleε1 1/vextendsingle/vextendsingle= 1−5π 16/similarequal0.018/lessmuch1, p≤max/vextendsingle/vextendsingleε1 n/vextendsingle/vextendsingle=/vextendsingle/vextendsingleε1 1/vextendsingle/vextendsingle=/vextendsingle/vextendsingle1−21π 64/vextendsingle/vextendsingle/similarequal0.031/lessmuch1.(3. 27) (This estimate follows from the observation that the matrix operator with components ˆQ(l,−1 2) nm is a projection operator of norm at most unity.) Thus, the systems (3 .24) and (3.26) can be solved very eﬀectively by the method of ©200 1 CRC Press LLC successi veapproximation sforanyvalueoftheparamete ru1(orθ1).Appr ox- imateanalytica lexpression sforcapacitanc egivenattheendofthissection arederivedinthisway. Letusnowturnattentiontothecharge dspherica lbarrel .Assum ethatthis doubly-connecte dconducto rischarge dtounitpotential,i.e., U(a,θ)=1,θ∈(θ0,π−θ0). (3.28) Followingasimila rprocedur etotheaboveproducesthedualseriesequations   ∞/summationtext n=0/parenleftbig n+1 4/parenrightbig bnP(−1 2,0) n (u)=0,u∈(−1,u1) ∞/summationtext n=0bnP(−1 2,0) n (u)=1,u∈(u1,1)(3.29) wherebn=(−1)nx2nandu1=−u0(θ1=π 2−θ1).Apreliminar yintegration isnecessar ytotransfor mtheseequation stothestandar dformconsidere din Sectio n2.1. ∞/summationdisplay n=1n+1 4 nbnP(1 2,1) n−1(u)=√ 2−(1−u)1 2 (1−u)1 2(1+u)b0,u∈(−1,u1)(3.30) ∞/summationdisplay n=1bn nP(1 2,1) n−1(u)=4 1+u(1−b0),u∈(u1,1)(3.31) Theﬁnalforma tofthesolutio nisdeduce dfromtheresult sofSectio n2.1; omitting details, it is ds−∞/summationdisplay n=1dnµn/braceleftBigg ˆQ(1,1 2) n−1,m−1(u1) +2√ 2 α(u1)Qn(u1)Qm(u1)/bracerightBigg =2√ 2 α(u1)Qs(u1),(3. 32) wheres= 1,2,...; the coeﬃcient b0is computed from the formula b0= (α(u1))−1/braceleftBigg 1 +∞/summationdisplay n=1dnµnQn(u1)/bracerightBigg (3. 33) Furthermore, dn=/parenleftbigg n+1 4/parenrightbigg h(0,3 2) n−1/braceleftbigg h(1,1 2) n−1/bracerightbigg−1 2Γ (n+ 1) Γ/parenleftbig n+3 2/parenrightbigbn, µn= 1−n/parenleftbig n+1 2/parenrightbig n+1 4/bracketleftBigg Γ/parenleftbig n+1 2/parenrightbig Γ (n+ 1)/bracketrightBigg2 =O/parenleftbig n−2/parenrightbig asn→ ∞, (3. 34) ©200 1 CRC Press LLC α(u1) = 1−1 π/parenleftbigg1 +u1 2/parenrightbigg1 2 −1 2πln/bracketleftBigg 1−/radicalbig (1 +u1)/2 1 +/radicalbig (1 +u1)/2/bracketrightBigg , and Qn(u1) =1√π/parenleftbigg1 +u1 2/parenrightbigg3 2ˆP(0,3 2) n−1(u1)/radicalBig n/parenleftbig n+l 2/parenrightbig. The norm of the compact operator Hassociated with the system (3 .32) has the bound /bardblH/bardbl ≤max|µn|=µ1= 1−3π 10/similarequal0.057/lessmuch1; (3. 35) this estimate is uniform in the parameter u1. Hence, the solution of the system (3.32) is eﬀectively computed by means of successive approximations for any value of the parameter u1. We shall now calculate capacitances of these structures. The capacitance Cis related to the total charge qon a conductor at potential Uby q=CU. Thus, at unit potential, the capacitance Cnumerically coincides with value of the charge q. Charge is determined by integration of the surface charge densityσon the conductor surface; it is proportional to the jump in the normal component of the electric ﬁeld− →E= gradUon the conductor surface σ(θ) =1 4π{Er(a+ 0,θ)−Er(a−0,θ)}. (This follows from Equation (1. 2).) The concrete expression for σis σ(θ) =1 4πa∞/summationdisplay n=0(2n+ 1)xnPn(cosθ). (3. 36) 3.2.1 Approximate analytical formulae for capacitance Let us ﬁrst consider two caps at equal potential ( l= 0). The charge q1,1on each spherical cap is determined by integration of the function σ(θ) over the appropriate portion of the spherical surface r=a: q1,1=1 2ax0= 2−7 4√πab 0. (3. 37) From the trivial approximation/parenleftbig c0 2n= 0/parenrightbig one readily obtains from (3 .24) the approximation for c0: c(0) 0≈23 4√π1−ˆQ(0,−1 2) 00 (u1) 1−ε0 0ˆQ(0,−1 2) 00 (u1)=23 4√π.cosθ1 1−/parenleftbig 1−π 4/parenrightbig (1−cosθ1). (3. 38) ©200 1 CRC Press LLC θ1 0◦10◦20◦30◦40◦ a−1q(0) 1,10.5 0.49401 0.47600 0.44583 0.40326 a−1q(1) 1,10.5 0.49399 0.47583 0.44525 0.40220 θ1 50◦60◦70◦80◦90◦ a−1q(0) 1,10.34807 0.28004 0.19912 0.10554 0 a−1q(1) 1,10.34654 0.27835 0.19778 0.10498 0 Table 3.1 Appr oximat ecapacitance softhecharge dcappair. Substituting (3 .38) in (3.37) produces the approximation q1,1≈1 2acosθ1 1−/parenleftbig 1−π 4/parenrightbig (1−cosθ1). (3. 39) The simplest approximation for the capacitance of this pair of conductors is thus C(0) 1,1= 2q(0) 1,1=acosθ1 1−/parenleftbig 1−π 4/parenrightbig (1−cosθ1). (3. 40) It is worth noting that the same problem was solved in [42]. Despite obtain- ing a Fredholm integral equation of the second kind (which in itself does not guarantee solution eﬀectiveness), further analytical investigation was impossi-ble because the solution was highly dependent on the cap dimensions; only nu-merical results were obtained. Let us make some comparison of results (those of [42] are given in parentheses): when θ 1= 60◦, a−1q(0) 1,1= 0.280 (0.278) ; whenθ1= 30◦, a−1q(0) 1,1= 0.445 (0.445).Formula (3.40) is thus appealing in its simplicity and relatively good accuracy, demonstrating the advantages of the method presented here. The ﬁrst successive approximation provides a more accurate estimate of capacitance (or charge), and an approximate analytical expression for the po- tential distribution; we obtain the following approximation for the coeﬃcients: b(1) 2m=/parenleftBigg b0ε0 0−23 4√π/parenrightBigg (−1)mˆQ(0,−1 2) 0m (u1), (m= 1,2,...). (3. 41) In this approximation the charge is q(1) 1,1=1 2acosθ1−β(θ1) 1−/parenleftbig 1−π 4/parenrightbig {1−cosθ1+β(θ1)}(3. 42) where β(θ1) =∞/summationdisplay n=1ε0 nˆQ(0,−1 2) n0 (u1)ˆQ(0,−1 2) 0n (u1). (3. 43) ©200 1 CRC Press LLC Figure 3.2 Electrostatic potential near a pair of symmetrical spherical caps charged to unit potential with subtended angle θ0= 30o.Truncation numberNtr= 11. An approximation for the function βwith relative error not exceeding 3.10−4is β(θ1)≈5/summationdisplay n=1∆ε0 n[P2n−1(cosθ1)−P2n+1(cosθ1)]2/(4n+ 1) + 1 8/braceleftbigg −2 cosθ1(1−cosθ1) ln 2 +1 2sin2θ1−1 2(1−cosθ1)2ln (1−cosθ1)/bracerightbigg +1 8{2 cosθ1ln 2−cosθ1(1−cosθ1)−cosθ1(1 + cosθ1) ln (1 + cos θ1)} where ∆ε0 n=ε0n−1 16n(2n+ 1). (3. 44) Formulae (3 .40) and (3.42) were used to calculate q(0) 1,1andq(1) 1,1, respectively. ©200 1 CRC Press LLC−1.5 −1 −0.5 0 0.5 1 1.5−1.5−1−0.500.511.5 x/az/a 0.40.5 0.6 0.7 0.8 0.90.50.4 0.60.7 0.80.9 Figur e3.3 Electrostati cpotentialnearasymmetrica lpairofspherica lcaps charge dtounitpotentialwithsubtende dangleθ0=60o.Truncation numberNtr=11. Thevaluesofq(1) 1,1agreeperfectl ywithdatain[42].Somecompute dresults areprese ntedinTable3.1. Thespatia ldistributio nofthepotentialUcompute dfrom(3.14),after solutio nof(3.24)isdisplayedinFigure s3.2and3.3forthepairofcaps, atunitpotentialwithangleparamete rθ0=30◦andθ0=60◦,respectively. Itisevide ntthatmutualcouplin goftheelectrostati cﬁeldsproducedbythe smalle rpairofcharge dcapsissmall .Thelargerpairexhibit smuchstronger coupling ;theresulta ntﬁeldappearsnotasthecompositio noftwoindividual ﬁelds ,butasasingleelectrostati cﬁeldsurroundin gtheentirestructure. Furthermore ,theseﬁgure sillustrat ethatwell-separate dsmallcapsmight bereadil yanalyse dbyameth odofsuccessi veapproximations ,utilisin gthe knownpotentialofasingleisolate dcharge dcap.However,suchanapproa ch willfailforlargercaps(Figur e3.3);thechoiceofmeth odapplie discritical in producing an eﬃcient mathematical tool for analytical treatment of the ©200 1 CRC Press LLC−1.5 −1 −0.5 0 0.5 1 1.5−1.5−1−0.500.511.5 x/az/a 0.6 0.70.8 0.9 0.950.80.950.70.60.5 0.9 Figure 3.4 Electrostatic potential near a spherical condensor with subtended angleθ0= 30o.Truncation number Ntr= 11. problem. When oppositely charged ( l= 1 ), the caps form a capacitor or condensor. The charge on the lower cap is q−1,1=1 2a∞/summationdisplay n=0x2n+1[P2n(0)−P2n+2(0)] = 1 2a∞/summationdisplay n=0(−1)nc2n+1Γ/parenleftbigg n+1 2/parenrightbigg Γ (n+ 1)/bracketleftbigg h(−1 2,1) n/bracketrightbigg−1 2 [P2n(0)−P2n+2(0)].(3. 45) The ﬁrst approximation in solving Equations (3 .25) and (3.26) produces c(1) 1=−23 4/parenleftbigg3 π/parenrightbigg1 2 cosθ1 1 + (3π 8−1)(1−3 2cosθ1+1 2cos3θ1), ©200 1 CRC Press LLC−1.5 −1 −0.5 0 0.5 1 1.5−1.5−1−0.500.511.5 x/az/a −0.2 −0.4−0.6−0.80.20.40.6 0.8 Figure 3.5 Electrostatic potential near a spherical condensor with subtended angleθ0= 60o.Truncation number Ntr= 11. and c(1) 2n+1=c1ε1 0ˆQ(1,−1 2) 0n (u1)−2/braceleftBigg 1 +u1 π/parenleftbig n+1 2/parenrightbig (n+ 1)/bracerightBigg1 2 ˆP(0,1 2) n (u1).(3. 46) Substitution of these values in the formula (3 .45) yields an approximate ana- lytical expression for q−1,1: q(1) −1,1=−1 2a∞/summationdisplay n=1[P2n(0)−P2n+2(0)]2P2n+1(cosθ1) −9 8acosθ1/braceleftbigg 1−/parenleftbigg3π 8−1/parenrightbigg/parenleftbigg 1−3 2cosθ1+1 2cos3θ1/parenrightbigg/bracerightbigg−1 ×/braceleftBigg 1−2 3/parenleftbigg3π 8−1/parenrightbigg∞/summationdisplay n=1[P2n(0)−P2n+2(0)]2Vn(cosθ1)/bracerightBigg (3. 47) ©200 1 CRC Press LLC−1.5 −1 −0.5 0 0.5 1 1.5−1.5−1−0.500.511.5 x/az/a −0.4−0.6−0.8−0.20.20.40.6 0.8 Figur e3.6 Electrostati cpotentialnearaspherica lbarre lcharge dtounitpo- tentialwithapertur esubtendin gangleθ0=30o.Truncatio nnumber Ntr=11. where Vn(cosθ1)=−sin2θ1P2n+1(cosθ1) −2(4n+3)−1cosθ1[P2n+2(cosθ1)−P2n(cosθ1)] +2(4n+3)−1(4n+5)−1[P2n+3(cosθ1)−P2n+1(cosθ1)] −2(4n+3)−1(4n+1)−1[P2n+1(cosθ1)−P2n−1(cosθ1)]. Somecalculate dresult sarereproducedinTable3.2.Acompariso nofthe tabulate dresult swiththoseobtaine dbynumerica lsolutio nof(3.25)and (3.26)showsthatFormula(3.47)isaccurat etothreesigniﬁca ntdigits(over thewholerangeofθ1).Thespatia ldistributio nofthepotentialaroun dca- pacitor swithangleparamete rθ0=30◦and60◦areshowninFigure s3.4and ©200 1 CRC Press LLC−1.5 −1 −0.5 0 0.5 1 1.5−1.5−1−0.500.511.5 x/az/a 0.98 0.95 0.9 0.80.7 0.60.980.950.90.80.7 0.6 Figur e3.7 Electrostatic potential near a spherical barrel charged to unit po- tential with aperture subtending angle θ0= 60o.Truncation number Ntr= 11. 3.5,respectively.Thiswascompute dfrom(3.14)aftersolutio nof(3.26). Finally, we calculate the capacitance of the spherical barrel. The charge q1, and hence the capacitance of the doubly-connected spherical barrel conductorat unit potential, is determined by q 1=ab0.In the trivial approximation d(0) n= 0,and the corresponding estimate follows from (3 .32) : q(0) 1=c(0) 1=a/parenleftbigg 1−1 πcosθ1−1 2πln/bracketleftbigg1−cosθ1 1 + cosθ1/bracketrightbigg/parenrightbigg−1 . (3. 48) In the limiting case of free space ( θ1= 0), Formula (3 .48) produces the expected result that q(0) 1= 0.For the other limiting case of a closed spherical shell/parenleftbig θ1=π 2/parenrightbig ,it produces the expected result q(0) 1=a.A thin cylindrical ©200 1 CRC Press LLC−1.5 −1 −0.5 0 0.5 1 1.5−1.5−1−0.500.511.5 x/az/a0.95 0.9 0.9 0.950.80.70.60.5 0.8 0.7 0.6 0.5 Figur e3.8 Electrostati cpotentialnearaspherica lbarre lorringcharge dto unitpotentialwithapertur esubtendin gangleθ0=80o.Truncation numberNtr=11. ring(θ1/lessmuch1)hastheapproximat echarge q(0) 1∼=πa π−1+ln(2/θ1)∼=πa 0.07+ln(16/θ1) wherewehaveemployedtheapproximatio nπ−1∼=ln8+0.07.Thisisvery closetotheknownresultforthechargeonanarrowcylindrica lring[29].The estimat eofq1improveswiththenextapproximation .Sampl ecalculation sof q(0) 1arereproducedinTable3.3. Thedistributio noftheelectrostati cpotentialsurroundin gthreediﬀere ntly shapedbarrel s(θ0=30◦,60◦,and80◦)isdisplayedinFigure s3.6,3.7,and 3.8.Thiswascompute dfrom(3.14)aftersolvin g(3.32).Asmightbe expected, the potential is nearly constant inside the largest barrel. When the angleθ0= 80◦,the spherical barrel becomes a “ring.” ©200 1 CRC Press LLC−1.5 −1 −0.5 0 0.5 1 1.5−1.5−1−0.500.511.5 x/az/a 0.40.50.60.7 0.8 0.7 0.8 0.90.60.50.4 θ1 10◦20◦30◦40◦ a−1q(1) −1,1-1.729 -1.262 -0.967 -0.739 θ1 50◦60◦70◦80◦ a−1q(1) −1,1-0.549 -0.391 -0.241 -0.113 Table 3.2 T otalcharg eonthelowercapofthespherica lcondensor. θ1 10◦20◦30◦40◦50◦60◦70◦80◦ a−1q(0) 10.683 0.797 0.874 0.927 0.963 0.984 0.995 0.999 Table 3.3 Approximate capacitance of the spherical barrel as a function of angle θ1=π 2−θ0. 3.3Anasymmetricalpai rofsphericalcapsandtheasym- metricbarrel Inthepreviou ssectionweconsideredtwosphericalcapsthatsubtended equalanglesatth eoriginofth ecommonsphericalsurfaceonwhichthey bothlie.Retainingaxialsymmetr yoftheentir estructure ,wen owallowthe capstosubtenddiﬀerentangles ,αandπ−β,asshowninFigure3.9(a). Whenchargedtoconstantpotential,thestandardformulationofthi sbound- aryvalueproblemforLaplace’sequation producesth enonsymmetrictripl e seriesequation s(whi char esimilartoth esymmetric triple equations of the previous section)   ∞/summationtext n=0anPn(cosθ)=1,θ ∈(0,α) ∞/summationtext n=0(2n+ 1)anPn(cosθ)=0,θ ∈(α,β) ∞/summationtext n=0anPn(cosθ)=1,θ ∈(β,π).(3.49) Proceedin gasinSection2.7,wemay transform the Equations (3 .49) to the equivalent symmetric triple series equations ∞/summationdisplay n=0bnPn(x) =/braceleftbigg2ρ0ρ1 1 +ρ0ρ1+ (1−ρ0ρ1)x/bracerightbigg1 2 , x∈(−1,−x0)∪(x0,1) (3. 50) ∞/summationdisplay n=0(2n+ 1)bnPn(x) = 0, x ∈(−x0,x0). (3. 51) Here x0=ρ0−ρ1 ρ0+ρ1=sin ∆ sin ∆ 0, ©200 1 CRC Press LLC Figure 3.9 (a) An asymmetrical pair of spherical caps, (b) an asymmetric spherical barrel. where the parameter ∆ 0=1 2(α+β) is the angular coordinate of the middle of the slot and ∆ =1 2(β−α) is its semi-width. The right-hand side of (3 .50) has the Fourier-Legendre expansion /braceleftbigg2ρ0ρ1 1 +ρ0ρ1+ (1−ρ0ρ1)x/bracerightbigg1 2 =∞/summationdisplay n=0dnPn(x) (3. 52) where dn=/radicalbigg cos ∆−cos ∆ 0 cos ∆ + cos ∆ 0[1−q(∆,∆0)]qn(∆,∆0) (3. 53) and q(∆,∆0) =√cos2∆−cos2∆0−cos ∆ cos ∆ 0. In calculating the coeﬃcients dnwe used the integral 1/integraldisplay −1Ps(z)√ a+bzdz=1 (s+1 2)√ a+b[1−q(a,b)]qs(a,b), a>b (3. 54) which may be obtained from the Dirichlet-Mehler integral representation for the Legendre polynomials Pnand the tabulated deﬁnite integral [19] π/integraldisplay 0cos(sx) a+bcosxdx=π√ a2−b2/parenleftBigg√ a2−b2−a b/parenrightBiggs , a>b. In contrast to the symmetrical case where the ﬁnal solution requires only even or only odd coeﬃcients (according as the pair of shells are equally or ©200 1 CRC Press LLC/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 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/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1β aaz z ooa) b) α y xxyβα oppositelycharged),thesolutiontothenonsymmetricalstructurerequires botheven( l=0)an dodd(l=1)coeﬃcients.Th esystem saresol ved separatelyandtheresult sarecombined.Basedonresultsobtaine dinthe previoussection,th eﬁnalfor mofth esolutionis(for l=0,1): cl/bracketleftbigg 1−εl 0ˆQ(l,−1 2) 00 (u1)/bracketrightbigg =fl/bracketleftbigg 1−ˆQ(l,−1 2) 00 (u1)/bracketrightbigg + ∞/summationdisplay n=1/parenleftbig c2n+lεln−f2n+l/parenrightbigˆQ(l,−1 2) n0 (u1),(3.55) andwhe nm≥1, c2m+l−∞/summationdisplay n=1/parenleftbig c2n+lεln−f2n+l/parenrightbigˆQ(l,−1 2) nm (u1)= f2m+l+/parenleftbig clεl 0−fl/parenrightbigˆQ(l,−1 2) 0m (u1)(3.56) where u1=1−2sin2∆ sin2∆0, /braceleftbiggf2n+l c2n+l/bracerightbigg =(−1)nΓ(n+1) Γ/parenleftbig n+1 2/parenrightbig/braceleftbigg h(−1 2,l) n/bracerightbigg1 2/braceleftbiggd2n+l b2n+l/bracerightbigg ,n=0,1,2,..., andtheres tofthenotationcoincideswiththatusedintheprevioussection. Justasforequallysizedcaps,thisproblemmaybeeﬀectivelysolvedbythe methodofsuccessiveappr oximation sbecausethesameestimatesofth enorm givenby(3 .27)arevalid. Suppressingsomedetails,theelectrostaticpote ntialofthechargednon- symmetricalsphericalbarrel(displaye dinFigure3.9(b))alsolead stothe nonsymmetric triple series equations ∞/summationtext n=0(2n+ 1)anPn(cosθ) = 0, θ ∈(0,α)∪(β,π) ∞/summationtext n=0anPn(cosθ) = 1, θ ∈(α,β),(3. 57) which are converted in the usual way to the dual series equations ( l= 0,1) ∞/summationtext n=0(b2n+l−d2n+l)P(0,l−1 2) n (u) = 0, u ∈(−1,u0) ∞/summationtext n=0/parenleftbig n+1 2l+1 4/parenrightbig b2n+lP(0,l−1 2) n (u) = 0, u ∈(u0,1)(3. 58) whereu0= cos 2θ0.The standard solution process eventually yields a fast converging i.s.l.a.e. of the second kind for the Fourier coeﬃcients. The odd ©200 1 CRC Press LLC (l=1)indexsystemis Bm−∞/summationdisplay n=0BnτnˆQ(1 2,0) nm (u0)=Dm−∞/summationdisplay n=0DnˆQ(1 2,0) nm (u0)(3.59) where τn=1−/parenleftbigg n+3 4/parenrightbigg/bracketleftBigg Γ(n+1) Γ/parenleftbig n+3 2/parenrightbig/bracketrightBigg2 =O/parenleftbig n−2/parenrightbig asn→∞, and {Bn,Dn}=Γ/parenleftbig n+3 2/parenrightbig Γ(n+1)/braceleftbigg h(1 2,0) n/bracerightbigg1 2 {b2n+1,d2n+1}; thee ven(l=0)indexsystemis Gm−∞/summationdisplay n=1Gnµn/braceleftbigg ˆQ(1,1 2) n−1,m−1(u1)+2√ 2{γ(u1)}−1Qn(u1)Qm(u1)/bracerightbigg =2√ 2{γ(u1)}−1D0Qm(u1)+ ∞/summationdisplay n=1Dn/braceleftbigg ˆQ(1,1 2) n−1,m−1(u1)+2√ 2{γ(u1)}−1Qn(u1)Qm(u1)/bracerightbigg (3.60) whereγ(u1)≡α(u1), γ(u1)b0=D0+∞/summationdisplay n=1(Gnµn+Dn)Qn(u1), Gm=m+1 4 m+1 2Γ(m+1) Γ/parenleftbig m+1 2/parenrightbigh(0,3 2) m−1/braceleftbigg h(1,1 2) m−1/bracerightbigg−1 2 (−1)mb2m, and Dm=(−1)md2mΓ/parenleftbig m+1 2/parenrightbig Γ(m)h(0,3 2) m−1/braceleftbigg h(1,1 2) m−1/bracerightbigg−1 2 . Theremainingnotationcoincide swiththatwhichweusedinth esolutionof thesphericalbarrelwithequal-size daperture s(Section3.2). Some remarks about computation of the electrostatic ﬁelds are in order. It is not necessary to compute the original Fourier coeﬃcients {an}∞ n=0.Cal- culations can be done in terms of the secondary coeﬃcients {bn}∞ n=0. For instance, to derive formulae for capacitance and potential distribution along thez-axis, use Formula (2 .253) in which we set m= 0: an=√cos ∆ + cos ∆ 0 21/integraldisplay −1dx√cos ∆ + cos ∆ 0x ×Pn/parenleftbiggcos ∆ 0+ cos ∆x cos ∆ + cos ∆ 0x/parenrightbigg∞/summationdisplay s=0(2s+ 1)bsPs(x). (3. 61) ©200 1 CRC Press LLC Thetotalchargeaccumulate donbothcapsisQ=a.a0;from(3.54)oneﬁnds that Q=1 2a/radicalbig cos∆+cos∆0∞/summationdisplay s=0(2s+1)bs1/integraldisplay −1Ps(x)dx√cos∆+cos∆0x =a[1−q(∆,∆0)]∞/summationdisplay s=0bsqs(∆,∆0). (3.62) (Obser vethatforsymmetri ccaps∆=π 2,q/parenleftbig ∆,π 2/parenrightbig =0,andtheexpression (3.62)reduce stothepreviousl ystatedform,namel yQ=a.a0.) Theelectrostati cpotentialtakenalongthez-axis(sothatcosθ=±1)is givenby U(t,±1)=∞/summationdisplay n=0an(±t)n, (3.63) wheret=r/a≤1.Uponsubstitutin g(3.61)andtakin gaccou ntoftheseries ∞/summationdisplay n=0Pn/parenleftbiggcos∆0+cos∆x cos∆+cos∆0x/parenrightbigg (±t)n=/parenleftbigg 1∓2tcos∆0+cos∆x cos∆+cos∆0x+t2/parenrightbigg−1 2 (derivedfromthegeneratin gfunctio nforPn,seeAppendix ,(B.59)),and thevalueoftheintegra lgivenby(3.54),weobtai ntheﬁnalformulaforthe distributio noftheelectrostati cpotentialalongthez-axisintermsofthe coeﬃcie ntsbn: U(t,±1)=1 1∓t[1−R(∆,∆0;t)]∞/summationdisplay s=0bsRs(∆,∆0;t), (3.64) where R(∆,∆0;t)=/parenleftbig 1−t2/parenrightbig√cos2∆−cos2∆0−/parenleftbig 1+t2/parenrightbig cos∆±2tcos∆0 (1+t2)cos∆0∓2tcos∆. (3.65) Whenr>a,weusetheformula U(ρ,±1)=∞/summationdisplay n=0an(±ρ)−n−1=t∞/summationdisplay n=0an(±t)n, (3.66) whereρ=t−1=r/a>1,sothattheexpressio n(3.64)maybeemployed. Notethatwiththelimitin gvaluest=0,1wehave R(∆,∆0;0)=q(∆,∆0),R(∆,∆0;1)=±1. Somecalculation softhetotalchargeonspherica lcapsofunequa lsizeare displayedinTable3.4,andthesimila rcalculation sfornonsymmetrica lspher- icalbarrel saredisplayedinTable3.5.Anillustrati veexampl eoftheelec- trostati cpotentialdistributio nalongthez-axisforanasymmetrica lpairof spherica lcapsareshowninFigur e3.10. ©200 1 CRC Press LLC ∆/downslope∆075◦60◦45◦30◦ 0◦1 1 1 1 5◦0.99710 0.99742 0.99788 0.99850 10◦0.98839 0.98959 0.99148 0.99405 15◦0.97381 0.97647 0.98076 0.98684 20◦0.95327 0.95800 0.96568 0.97751 25◦0.92673 0.93409 0.94625 0.96776 30◦0.89413 0.90469 0.92265 — 45◦0.75968 0.78368 — — 60◦0.57036 — — — Normalise dtotalcharg eontwononsymmetrica lspherica lcapsa−1Q1,1. ∆ 0 istheangula rcoordinat eofthemiddl eoftheslot,∆isitssemiwidth. ∆/downslope∆075◦60◦45◦30◦ 0◦0 0 0 0 5◦0.58555 0.53669 0.45724 0.34964 10◦0.67380 0.62062 0.53415 0.41722 15◦0.73665 0.68163 0.59228 0.47193 20◦0.78640 0.73100 0.64119 0.52084 25◦0.82735 0.77264 0.68415 0.56621 30◦0.86158 0.80843 0.72264 — 45◦0.93426 0.88970 — — 60◦0.97410 — — — Normalise dtotalcharg eonanonsymmetrica lspherica lbarre la−1Q1,1. ∆ 0 istheangula rcoordinat eofthemiddl eoftheslot,∆isitssemiwidth. ©200 1 CRC Press LLCTable 3.5Table 3.4 Figure 3.10 Electrostatic potential along the z-axis for an asymmetrical pair of spherical caps charged to unit potential and subtending angles α andπ−β. 3.4 The method of inversion The method of inversion in a sphere is described in many classical books on electromagnetism (see for example [54], [66]). In three-dimensional potential (electrostatic) problems this method plays, to some extent, the same role asconformal mapping does in two-dimensional problems. It is mainly used inthe calculation of capacitance of closed charged shells. To this end, let usstate a very useful theorem ﬁrst formulated by C. J. Bouwkamp [7] in 1958. Theorem 5 Consider an isolated (or solitary) conductor bounded by a sur- faceS. LetS/primebe the image of Sunder inversion in the sphere of radius a, centred at a given ﬁxed point M. LetU 0be the free-space potential due to a unit negative charge located at M.LetU0+U1be the total potential in- ©200 1 CRC Press LLC−2 −1.5 −1 −0.5 0 0.5 1 1.5 20.20.30.40.50.60.70.80.911.11.2 z / aU α = 150, β = 1350 α = 300, β = 1350 α = 450, β = 1350 α = 450, β = 1650 ducedbythisuni tcha rgeatMinthepresenceof S/primewhenitisgrounded(i.e., U0+U1=0 onS/prime).IfV0isthevalueoftheindu cedpotential U1atM/parenleftbig V0=U1(M)/parenrightbig ,thenthecapacitance Coftheconductor Sequal sa2V0. Weintroducetwowell-knownexamplestoillustratetheus eofthistheorem inthesimples tcases.Theﬁrs texampl ecalculatesthecapacitanceofasingle sphericalcap.Thesecond ,borrowe dfrom[7],calculate sthecapacitanceof twotou chingsphericalshells. Wehavealreadycalculate dthecapacitanc eCcapofth esphericalcapin Section1.4: Ccap=a.a0,wher eaisradiusofthespher eanda0islowest Fouriercoeﬃcientoftheexpansionofth eelectrostati cpote ntialinFourier- Legendreseries ;thusa0=π−1(θ0+sinθ0),an dCcap=aπ−1(θ0+sinθ0). Letusdemonstrat eanalternativewayofarrivin gatthisresultvi ainversion. Considerthesphericalcapsubtendin ganangleθ0attheoriginassh ownin Figure3.11.Itoccupiestheregion0 ≤θ≤θ0ofthesphericalsurface r=a. Beforeperformin ganinversionaboutthecentre Mlocatedatr=a,θ=0, werelocatethe caps othatitoccupie stheareaπ−θ0≤θ≤πonth esurfac e r=a.Under aninversioninth espher eofce ntreMandinversionradius R=2a,thesphericalcapistransforme dtothecirculardis cshownwith centreO/prime.Thecapacitancecalculation fo rasphericalcapi stransforme dto theequivale ntcalculationofth epotential Uforthegrounde dcirculardiscof radiusbinthepresenc eoftheunitnegativecharge,whichisplace datth e centreofinversion . LetO/primebeth eoriginofacylindricalpolarcoordinate system ( ρ,z),so that the coordinates of the inversion centre Mareρ= 0, z= 2a; the inversion procedure described above is given by ρ=Rtan1 2θ,andtheradius ofthe circulardiscimage is b=Rtan1 2θ0.Thepotentialfunctionemanatingfrom thenegativeunitchargei sU0=−/parenleftbig ρ2+z2/parenrightbig−1 2.Bythemethodo fseparation ofvariables ,wemaysee ktheaxisymmetricelectrostaticpote ntialU≡U(ρ,z) asth esumU=U0+U1,wheretheinduced potential U1has theform U1=/integraldisplay∞ 0f(ν)J0(νρ)e−ν|z−a|dν (3.67) andtheunknownfunction fistobedetermined .Uponenforcin gthemixe d boundaryconditionsonereadilyobtains thefollowin gdualserie sequations, involvin gBessel functions: /integraldisplay∞ 0f(ν)J0(νρ)dν=/parenleftbig ρ2+ 4a2/parenrightbig−1 2, 0≤ρ<b, (3. 68) /integraldisplay∞ 0νf(ν)J0(νρ)dν=0,ρ>b. WemayusetheresultsofSection2.6toﬁnd f(ν) =4a π/integraldisplayb 0cos(νρ) ρ2+4a2dρ, (3. 69) ©200 1 CRC Press LLC Figur e3.11 Thespherica lcapanditsimag e(thecircula rdisc)unde rtheinver- sionprocedur e(seetext). b=2atan1 2θ0.Accordin gtoBouwkamp’ stheorem ,thecapacitanc eis Ccap=R2U1(M)=4a2/integraldisplay∞ 0f(ν)e−2νadν. (3.70) Thus,thecapacitanc eequals Ccap=4 πa3/integraldisplay∞ 0dνe−2νa/integraldisplayb 0dρcos(νρ) ρ2+4a2=32 πa4/integraldisplayb 0dρ (ρ2+4a2)2, andaneleme ntarycalculatio nleadsto Ccap=a π(θ0+sinθ0), whichisinaccor dwiththepreviou sresult. Oursecon dexampl eisthecalculatio nofcapacitanc eoftwotouchingspheres ofradiiaandb,a≤b(Figur e3.12).In[7],thisproble mwastreate dbythe image method. With a view to extending it to open touching spherical shells, we derive a solution by the method of separation of variables. The inversionsphere has centre at the point of contact Mand radius 2 b.LetMbe the origin of polar cylindrical coordinates. The transformation (given by ρ= 2atan 1 2θ) ©200 1 CRC Press LLC/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1M y xo y’ x’o’aθ0z , z’ b Figure 3.12 Touching spheres of radii a,b. transforms the electrostatic problem for two touching spheres that are charged to unit potential to the equivalent electrostatic calculation for two grounded inﬁnite planes, separated by a distance l= 2b(1 +b/a),in the presence of a unit negative charge at inversion centre M. As before we may seek a solution in the form U(ρ,z) =U0+U1, (3. 71) where U0=−/parenleftbig ρ2+z2/parenrightbig−1 2, (3. 72) U1=/integraldisplay∞ 0/braceleftbig f(ν)e−νz+g(ν)eνz/bracerightbig J0(νρ)dν, (3. 73) withfandgto be determined. Notice U(ρ,z)→0 asρ→ ∞.The boundary conditionsU(ρ,−2b) =U/parenleftbig ρ,2b2/a/parenrightbig = 0 (each plane is grounded) imply /integraldisplay∞ 0/braceleftbig f(ν)e2νb+g(ν)e−2νb/bracerightbig J0(νρ)dν=/parenleftbig ρ2+ 4b2/parenrightbig−1 2, 0<ρ< ∞,(3. 74) /integraldisplay∞ 0/braceleftBig f(ν)e−2νb2/a+g(ν)e2νb2/a/bracerightBig J0(νρ)dν=/parenleftBig ρ2+ 4b2(b/a)2/parenrightBig−1 2, 0<ρ< ∞.(3. 75) ©200 1 CRC Press LLCM ba2b /a 2b2Ψ=0 Ψ=0 ABesselintegraltransform,applie dtoequations(3 .74)and(3.75)shows that f(ν)=sinh/parenleftbig 2νb2/a/parenrightbig sinh(2νb(a+b)/a)e−2νb, (3.76) g(ν)=sinh(2νb) sinh(2νb(a+b)/a)e−2νb2/a. (3.77) Bouwkamp’stheoremimpliesthatthecapacitanceofth etwotouchin gspheres is Ca,b=4b2U1(M)=4b2/integraldisplay∞ 0{f(ν)+g(ν)}dν (3.78) =−ab a+b/braceleftbigg 2γ+ψ/parenleftbigga a+b/parenrightbigg +ψ/parenleftbiggb a+b/parenrightbigg/bracerightbigg , (3.79) whereγisEuler’ sconstant,and ψdenotesth elogarithmicderivativeofthe GammafunctionΓ(see[1]), ψ(−x)=−γ+x−1−x∞/summationdisplay n=11 n(n−x). Whenthesphericalradiiareequal( a=b),ψ(1 2)=−γ−2ln2,andthe capacitance Cb,bequals2bln2. Letu sextendth elastexampletoconsideropensphericalcaps;various conﬁgurationsar eshowninFigure3.13.Werestrictoursel vestospheresof equal radiia, and shells subtending equal angles θ0,and concentrate on the last two conﬁgurations (c) and (d); the solution to the ﬁrst two is easily deduced from the last two (using image theory). From the symmetry after inversion, it is suﬃcient to consider the problem in the upper half-space ( z≥0). Thus, we ﬁnd the distribution of the electrostatic potential UinR3,which is due to the unit negative charge located at inversion centre Min presence of two grounded circular discs, separated by a distance l= 4a. As before, the free-space potential emanating from the negative unit charge isU0=−/parenleftbig ρ2+z2/parenrightbig−1 2.Subdivide the space into two regions. In region I , 0<z≤2a,we seek a solution in the form UI=U0+U(i)(3. 80) where U(i)=/integraldisplay∞ 0f(ν)J0(νρ) cosh (νz)dν; (3. 81) in region II , z>2a,we seek a solution in the form UII=U0+U(e)(3. 82) ©200 1 CRC Press LLC Figure 3.13 Various conﬁgurations of spherical cap pairs. where U(e)=/integraldisplay∞ 0g(ν)J0(νρ)e−νzdν (3. 83) and the functions f,gare to be determined. (The form of U(i)andU(e)is a superposition of partial solutions to Laplace’s equation, which vanish at inﬁnity.) From the continuity condition UI(ρ,2a) =UII(ρ,2a),0≤ρ<∞ we deduce cosh (2νa)f(ν) =e−2νag(ν). (3. 84) The mixed boundary conditions applied on the plane z= 2agive U(i)(ρ,2a) =U(e)(ρ,2a) =−U(0)(ρ,2a), 0≤ρ<b, (3. 85) ∂U(i) ∂z(ρ,2a) =∂U(e) ∂z(ρ,2a), ρ>b, (3. 86) whereb= 2atan1 2θ0.We therefore obtain the following dual integral equa- tions for the unknown function f: /integraldisplay∞ 0f(ν) cosh (2νa)J0(νρ)dν=/parenleftbig ρ2+ 4a2/parenrightbig−1 2,0≤ρ<b, (3. 87) /integraldisplay∞ 0νf(ν)e2νaJ0(νρ)dν= 0, ρ>b. (3. 88) ©200 1 CRC Press LLC/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 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/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1M ao o’z a) b) c) d) zzz oo oo’ o’ o’MMMθ θθ θ θ θ0 00 00 0 It is convenient to introduce a new unknown function Fby F(ν) =e2νaf(ν), (3. 89) and transform the dual equations to the weighted form /integraldisplay∞ 0/parenleftbig 1 +e−4νa/parenrightbig F(ν)J0(νρ)dν= 2/parenleftbig ρ2+ 4a2/parenrightbig−1 2,0≤ρ<b, (3. 90) /integraldisplay∞ 0νF(ν)J0(νρ)dν= 0, ρ>b. (3. 91) Following the Abel integral transform technique, these equations produce /integraldisplay∞ 0ν1 2F(ν)J−1 2(νρ)dν =/parenleftbigg2 π/parenrightbigg1 24aρ−1 2 ρ2+ 4a2−/integraldisplay∞ 0ν1 2F(ν)e−4νaJ−1 2(νρ)dν, ρ<b, (3. 92) /integraldisplay∞ 0ν1 2F(ν)J−1 2(νρ)dν= 0, ρ>b. (3. 93) Application of the Bessel-Fourier integral transform to both parts of this equation produces a Fredholm integral equation of the second kind. From a computational point of view, however, the discrete form of solution is prefer-able. To reduce (3 .92) and (3.93) to an i.s.l.a.e., we use the Hankel transform to obtain µ −1 2F(µ) = 4a/parenleftbigg2 π/parenrightbigg1 2/integraldisplayb 0ρ1 2J−1 2(µρ) ρ2+ 4a2dρ −/integraldisplayb 0ρJ−1 2(µρ)/braceleftbigg/integraldisplay∞ 0ν1 2F(ν)e−4νaJ−1 2(νρ)dν/bracerightbigg dρ(3. 94) and then represent unknown function Fby a Neumann series F(µ) =/parenleftbigg2 πbµ/parenrightbigg1 2∞/summationdisplay n=0(4n+ 1)1 2xnJ2n+1 2(µb) (3. 95) where it can be shown that {xn}∞ n=0∈l2. Substitute (3 .95) into (3.94). Then multiply both sides of (3 .94) by (4m+ 1) 1 2J2m+1 2(µb), integrate over [0 ,∞),and use the well-known integral formula [19], /integraldisplay∞ 0t−1Jν+2n+1(t)Jν+2m+1(t)dt= (4n+ 2ν+ 2)−1δnm. (3. 96) ©200 1 CRC Press LLC This yields an i.s.l.a.e. of the second kind for the coeﬃcients {xn}∞ n=0, xm+∞/summationdisplay n=0αnmxn=βm, (3. 97) wherem= 0,1,2,...,and αnm= [(4n+ 1) (4m+ 1)]1 2/integraldisplay∞ 0ν−1e−4aνJ2n+1 2(νb)J2m+1 2(νb)dν,(3. 98) βm= 2 tanθ0 2(−1)m(4m+ 1)1 2/integraldisplay1 0P2m(t) 1 +t2tan21 2θ0dt. (3. 99) Let us determine the capacitance Cof two spherical caps in terms of the Fourier coeﬃcients xn.As before, C= 4a2/integraldisplay∞ 0f(ν)dν= 4a2/integraldisplay∞ 0F(ν)e−2νadν, (3. 100) so substituting for Ffrom (3.95),we ﬁnally deduce that the capacitance C equals 2a√π∞/summationdisplay n=0xn(4n+ 1)1 2Γ (2n+ 1) Γ/parenleftbig 2n+3 2/parenrightbigtan2n1 2θ0 22n× 2F1/parenleftbigg n+1 2,n+ 1; 2n+3 2;−tan2θ0 2/parenrightbigg .(3. 101) Both Formulae (3 .100) and (3 .101) are valid for θ0<π 2.For small caps (θ0/lessmuch1),one can deduce approximate analytical expressions for capacitance in powers of the small parameter ε= tan1 2θ0/lessmuch1.To estimate of their accuracy, we express αnmas a hypergeometric function by direct calculation [14] of the integral in (3 .98): αnm=[(4n+ 1) (4m+ 1)]1 2 24n+4m+2/parenleftbigg tanθ0 2/parenrightbigg2n+2m+1Γ (2n+ 2m+ 1) Γ/parenleftbig 2n+3 2/parenrightbig Γ/parenleftbig 2m+3 2/parenrightbig ×4F3/parenleftbigg p,p+1 2,p−1 2,p; 2p,2n+3 2,2m+3 2;−tan2θ0 2/parenrightbigg (3. 102) wherep=n+m+ 1.Also we may calculate from (3 .99) using the tabulated integral [14], that βm= (4m+ 1)1 2Γ (m+ 1) Γ/parenleftbig m+1 2/parenrightbig Γ/parenleftbig 2m+3 2/parenrightbig tan2m+1θ0 2× 2F1/parenleftbigg m+1 2,m+ 1; 2m+3 2;−tan2θ0 2/parenrightbigg .(3. 103) ©200 1 CRC Press LLC Ifε=tan1 2θ0/lessmuch1,then βm=(4m+1)1 2ε2m+1Γ/parenleftbig m+1 2/parenrightbig Γ(m+1) Γ/parenleftbig 2m+3 2/parenrightbig× /braceleftBigg 1−/parenleftbig m+1 2/parenrightbig (m+1) 2m+3 2ε2+O/parenleftbig ε4/parenrightbig/bracerightBigg .(3.104) Wemayn owapplyth emeth odofsuccessiveapproximation sto(3.97): x(i+1) m =βm−∞/summationdisplay n=0αnmx(i) n, (3.105) wherei=0,1,...,andx(0) m=0.So x(1) m=βm, x(2) m=βm−∞/summationdisplay n=0αnmx(1)n=βm−∞/summationdisplay n=0αnmβm, ands oon(form=0,1,...). From(3.101),itcanbereadil yshownthataccurac yoforder O/parenleftbig ε2/parenrightbig is obtainedfor x0byneglectin gtherestofFouriercoeﬃcients xn(n≥1).Thus, since x(1) 0=2ε+O/parenleftbig ε3/parenrightbig , (3.106) x(2)0=2ε/parenleftbigg 1−1 πε/parenrightbigg +O/parenleftbig ε3/parenrightbig , anapproximat eformulaforcapacitanceis C=4a πx(2)0+O/parenleftbig ε3/parenrightbig (3.107) sothatthecapacitanceoftwosphericalcapsisapproximately C=4θ0 π/parenleftbigg 1−1 2πθ0/parenrightbigg +O/parenleftbig θ3 0/parenrightbig . (3.108) Thisformulahasaclearphysicali nterpretation.Theﬁrs ttermisthesu mof thecapacitancesoftwoisolatedsphericalcaps.Thesecondquadrati cterm reﬂectstheinteractionormutualimpactofthecaps. Thecapacitanceofth estructuresh owninFigure3.13(d )isobtaine dina similar way. This approach can be extended to consider spherical shells of diﬀering radii and angle. ©200 1 CRC Press LLC Figur e3.14 Spherically-sha pedelectroni clens. 3.5Electrostati cﬁeldsinaspherica lelectroni clens Inthissectio nanothe rillustratio nofmeth odsdevelopedforapplication sin aspherica lgeometr ycontextisgiven.Wecalculat etheelectrostati cﬁeldof aspherically-sha pedelectroni clens,showninFigur e3.14.Thespherically- shaped lens is a variant of a widely used electronic lens that comprises two charged, ﬁnite hollow cylinders at diﬀerent potentials V1andV2,aligned along a common axis of rotational symmetry. The upper electrode is the spherical shell segment given by r=a,θ 0≤θ≤π 2−δ; the lower electrode is its mirror image in the xy-plane. The distance between electrodes is negligibly small compared with the electrode dimension ( δ≈0), so that we model the lens by closely adjoined electrodes, electrically isolated by an inﬁnitesimally thin layer of dielectric. Let the upper electrode be charged to potential V1and the lower one charged to potential V2.Due to the rotational symmetry of the problem we seek the electrostatic potential V=V(r,θ) as an expansion in a Fourier-Legendre ©200 1 CRC Press LLC/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0 /1/1/1/1/1/1 /0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/0/0/0/0 /1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0 /1/1/1/1/1/1/0/0/0/0/0/0 /1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1z V V1 2θ θ00 o xy series(cf .(3.14)) V=∞/summationdisplay n=0xnPn(cosθ)/braceleftbigg(r/a)n,r<a (r/a)−n−1,r>a/bracerightbigg . (3.109) Useofthemixedboundarycondition satr=aandofsymmetr y,producesthe followingdecoupleddualseriesequation sfortheevenan doddindexFourier coeﬃcie nts:   ∞/summationtext n=0(n+1 4)x2nP2n(cosθ)=0,θ∈(0,θ0) ∞/summationtext n=0x2nP2n(cosθ)=1 2(V1+V2),θ∈/parenleftbig θ0,π 2/parenrightbig. (3.110)   ∞/summationtext n=0(n+3 4)x2n+1P2n+1(cosθ)=0,θ∈(0,θ0) ∞/summationtext n=0x2n+1P2n+1(cosθ)=1 2(V1−V2),θ∈/parenleftbig θ0,π 2/parenrightbig (3.111) TheﬁrstpairofEquation s(3.110)haveessentiallybeensolvedinSection 3.2,andm aybeidentiﬁe dwithEquation s(3.29)onceweset bn=(−1)nx2n; thesolutio ngive nby(3.32)mustbemultipliedbyafactor1 2(V1+V2). Thete chniquedevelopedinChapte r2maybefollowe dtoreduc ethesecond pair of Equations (3 .111) to the following i.s.l.a.e. with a matrix operator that is a completely continuous perturbation of the identity (in l2). Temporarily, we replace the right-hand side of the second equation in (3 .111) by unity, so that in the ﬁnal solution each Fourier coeﬃcient must be multiplied by a factor1 2(V1−V2): (1−pm)X2m+1+∞/summationdisplay n=0X2n+1pnˆQ(0,1 2) nm(u1) =Am, (3. 112) wherem= 0,1,2,..., and pn= 1−/parenleftbigg n+3 4/parenrightbigg/bracketleftBigg Γ (n+ 1) Γ/parenleftbig n+3 2/parenrightbig/bracketrightBigg2 =O/parenleftbig n−2/parenrightbig asn→ ∞, x2n+1= (−1)nΓ (n+ 1) Γ/parenleftbig n+3 2/parenrightbig/braceleftbigg h(0,1 2) n/bracerightbigg−1 2 X2n+1, (3. 113) and Am=/parenleftbigg2 π/parenrightbigg1 21/integraldisplay u1(1−u)−1 2(1 +u)1 2ˆP(0,1 2) m (u)du. ©200 1 CRC Press LLC An approximate analytical formula for the electrostatic potential along the axis of an electronic lens may be deduced. Set θ=/braceleftbigg 0 π/bracerightbigg in (3.109), and let q=r/a(r<a ) so that V/parenleftbigg q,0 π/parenrightbigg =1 2(V1+V2)∞/summationdisplay n=0x2nq2n±1 2(V1−V2)q∞/summationdisplay n=0x2n+1q2n(3. 114) (the plus and minus signs are associated with 0 and π,respectively). Then, using the approximate analytical solution for even and odd Fourier coeﬃcients (see (3. 32) and (3. 33)), x0/similarequal(α(u1))−1, x2n/similarequal(α(u1))−1(−1)n √πΓ/parenleftbig n+1 2/parenrightbig Γ (n+ 1)/parenleftbigg1 +u1 2/parenrightbigg3 2 P(0,3 2) n−1(u1), (3. 115) and x2n+1/similarequal(−1)n √πΓ/parenleftbig n+3 2/parenrightbig Γ (n+ 1)1/integraldisplay u1(1−u)−1 2(1 +u)1 2P(0,1 2) n (u)du. (3. 116) Substituting in the formula (3 .114) we obtain V/parenleftbigg q,0 π/parenrightbigg = (V1+V2) 2α(u1)/braceleftBigg 1 +1√π/parenleftbigg1 +u1 2/parenrightbigg3 2∞/summationdisplay n=1(−1)nΓ/parenleftbig n+1 2/parenrightbig Γ (n+ 1)q2nP(0,3 2) n−1(u1)/bracerightBigg ±(V1−V2)q 2√π∞/summationdisplay n=0(−1)nΓ/parenleftbig n+3 2/parenrightbig Γ (n+ 1)q2n1/integraldisplay u1(1 +u)1 2 (1−u)1 2P(0,1 2) n (u)du(3. 117) The integral contained in (3 .117) is tabulated in [14] so that (if u1=−u0) 1/integraldisplay u1(1 +u)1 2 (1−u)1 2P(0,1 2) n (u)du = (−1)nu0/integraldisplay −1(1−v)1 2 (1 +v)1 2P(1 2,0) n (v)dv = 2 (1 −u1)1 23F2/parenleftbigg −n−1 2,n+ 1,1 2; 1,3 2;1−u1 2/parenrightbigg .(3. 118) ©200 1 CRC Press LLC Sinceq<1,we may change the order of summation and integration in the last term of (3 .117) and so are led to the series also tabulated in [14], S(q,u) =∞/summationdisplay n=0Γ/parenleftbig n+3 2/parenrightbig Γ (n+ 1)/parenleftbig −q2/parenrightbignP(0,1 2) n (u) =∞/summationdisplay n=0Γ/parenleftbig n+3 2/parenrightbig Γ (n+ 1)/parenleftbig q2/parenrightbignP(1 2,0) n (−u) =√π 2/parenleftbig 1 +q2/parenrightbig−3 22F1/parenleftbigg3 4,5 4; 1;2q2 (1 +q2)2(1 +u)/parenrightbigg .(3. 119) This completes the derivation of an approximate formula for the potential distribution along the axis. Note at once that the value of the electrostatic potential at the origin ( z= 0) is V/parenleftbigg 0,0 π/parenrightbigg ∼=1 2(V1+V2)/braceleftbigg 1−1 πcosθ1−1 πln/parenleftbigg tanθ1 2/parenrightbigg/bracerightbigg−1 ; (3. 120) it is uniformly valid with respect to the parameter θ1∈/parenleftbig 0,π 2/parenrightbig . Further approximate analytical expressions which are uniformly valid with respect to the electrode dimensions, are rather complicated except for the lim- iting case of short electrodes ( |u1−1| /lessmuch1 orθ1/lessmuch1). A crude approximation to the electrostatic ﬁeld for narrow or very short electrodes is V/parenleftbigg q,0 π/parenrightbigg ∼=1 2(V1+V2)/parenleftbig 1 +q2/parenrightbig−1 2π ln (2/θ1)+O(θ1). (3. 121) For general lens parameters, numerical calculations may be simply and satisfactorily performed. If a truncation number Ntrof 6 to 8 is used to solve systems (3.112), (3.32), and (3.33), at least four signiﬁcant digits in the values of Fourier coeﬃcients X2n,X2n+1can be obtained stably. 3.6 Frozen magnetic ﬁelds inside superconducting shells In contrast to previous sections, we now consider a physical problem that mathematically reduces to a Neumann problem. The physical situation con- cerns a spherical thin shell with two symmetrically located circular holes(“doubly-connected” in a topological sense), manufactured from supercon- ducting material with critical temperature T c.Suppose this material is a su- perconductor of the ﬁrst kind so that when T >T cthis material behaves as normal metal, but when T≤Tc,it behaves as a superconductor. Place this shell (atT > T c) in some region of space that is permeated by a homoge- neous magnetic ﬁeld. Cool the shell in order to make the transition to the ©200 1 CRC Press LLC superconductingstate( T≤Tc),and switchoﬀ themagneticﬁeld .Assuming aperfec t(ideal)Meissnereﬀect,th emagneti cﬂuxΦ= πa2H0isfrozen in theshell’sc avity.Thedesignofspecialmagneticﬁeldcompressorsthatraises thethresholdsensitivityofsu perconductin gmagneticsystemsexploitsthis principle. Amathematicalanalysisofthisphenomenonrequire sthesolutionofa mixedboundary-valueproblemforthemagnetostaticpotential Um(r,θ)with aNeumannboundaryconditiongive nonth eshell’ ssurface.I naddition ,the frozen magneticﬂu xmus ttakeconstantvalu ethroughanyarbitrarilytaken cross-sectionofth eshell,includingacontou ronth esurfac eoftheshell. ConsideringLaplace’sequation ,togetherwithth econtinui tyconditionfor thenormalderivativeof Umatr=aandtheO/parenleftbig r−1/parenrightbig behaviourofthe potentialatinﬁni ty(r→∞ ),onem ayseekasolutioni ntheform Um(r,θ)=Φ πa∞/summationdisplay n=1AnPn(cosθ)/braceleftbigg(r/a)n,r<a −(n/(n+1))(r/a)−n−1,r>a/bracerightbigg , (3.122) whereΦ=πa2H0isthefrozen magneticﬂux, H0istheeﬀectivemeanvalu eof themagneticﬁeldtake natcross-section z=0,an d{An}∞ n=1aretheunkn own coeﬃcie ntstobedetermined ;theﬁnitenes sofenergycondition(seeSection 1.3)requires ∞/summationdisplay n=1|An|2<∞. Superconducting shells are usually modelled by ideal diamagnetic mate- rials of zero relative permeability; the normal component of magnetic ﬁeld vanishes at the shell surface. The boundary conditions on the potential are determined by continuity of radial and tangential components of the magnetic ﬁeld− →H=−gradUmon the superconducting portion of the shell (speciﬁed by the angular segment ( θ0,π−θ0)) and aperture, respectively: Hm r(a−0,θ) =Hm r(a+ 0,θ) = 0, θ∈(θ0,π−θ0), Hm θ(a−0,θ) =Hm θ(a+ 0,θ), θ∈(0,θ0)∪(π−θ0,π). The constancy of the magnetic ﬂux through any arbitrarily taken cross-section of the shell requires that if θ∈(θ0,π−θ0), 2πa2θ/integraldisplay 0Hm r(a,θ) sinθdθ= Φ. Applying these conditions to (3 .122),we obtain the following triple sym- ©200 1 CRC Press LLC metricequationsforth emodiﬁedFouriercoeﬃcie ntsxn=An/(n+1), ∞/summationdisplay n=1(2n+1)xnP1 n(cosθ)=0,θ∈(0,θ0)∪(π−θ0,π)(3.123) ∞/summationdisplay n=1xnP1 n(cosθ)=−1 2cosecθ,θ∈(θ0,π−θ0).(3.124) Becauseofthesymmetry, x2n≡0andthesetripleequationsar eequi valent tothedualpair ∞/summationdisplay n=0/parenleftbigg n+3 4/parenrightbigg x2n+1P1 2n+1(z)=0,z∈(−1,−z0)(3.125) ∞/summationdisplay n=0x2n+1P1 2n+1(z)=−1 2√ 1−z2,z∈(−z0,0)(3.126) wherez=cosθ,andz0=cosθ0. Aspreviouslydone ,(seeSection s3.2and3.3),weusethesubstitutions u=2z2−1and P1 2n+1(z)=√ 2(n+1 2)(1−u)1 2P(1,−1 2) n (u)(3.127) inEquation s(3.125)and(3 .126),andintegratethe mtoobtaindualseries equationswit hJacob ipolynomial sP(0,1 2) n, ∞/summationdisplay n=0x2n+1P(0,1 2) n (u)=2−3 2(1+u)−1 2ln/bracketleftBigg 1−/radicalbig (1+u)/2 1+/radicalbig (1+u)/2/bracketrightBigg ,u∈(−1,u0) (3.128) ∞/summationdisplay n=0/parenleftbigg n+3 4/parenrightbigg x2n+1P(0,1 2) n (u)=21 2(1+u)−1 2C,u∈(u0,1)(3.129) whereu0=2z2 0−1=cos2θ0,andCisanintegrationconstantdetermined byth econdition∞/summationtext n=1|An|2<∞. Equationssimilartothi sweresolvedinSection3.3;omittin gdetailsofits deduction, the ﬁnal system is X2m+1−∞/summationdisplay n=0X2n+1τnΠnm(u0) =Am, (3. 130) wherem= 0,1,2,..., and X2m+1= 21 4/bracketleftBigg/parenleftbig m+1 2/parenrightbig (m+ 1) m+3 4/bracketrightBigg1 2Γ/parenleftbig m+3 2/parenrightbig Γ (m+ 1)x2m+1. (3. 131) ©200 1 CRC Press LLC Furthermore, Πnm(u0)= ˆQ(−1 2,1) nm (u0)−√ 2Rn(u0)Rm(u0) ln/bracketleftBig/parenleftBig 1+((1 −u0)/2)1 2/parenrightBig //parenleftBig ((1+u0)/2)1 2/parenrightBig/bracketrightBig, Rs(u0)=/parenleftBigg 1−u0 2/parenleftbig s+1 2/parenrightbig (s+1)/parenrightBigg1 2 ˆP(1 2,0) s (u0), (3.132) Am=−2−3 2π1 2Rm(u0) ln/bracketleftBig/parenleftBig 1+((1 −u0)/2)1 2/parenrightBig //parenleftBig ((1+u0)/2)1 2/parenrightBig/bracketrightBig, and τn=1−/parenleftbigg n+3 4/parenrightbigg/bracketleftBigg Γ(n+1) Γ/parenleftbig n+3 2/parenrightbig/bracketrightBigg2 =O/parenleftbig n−2/parenrightbig ,asn→∞. Inthesam ewayasinSections3.2and3.3,thesyste m(3.130)hasan approximate analytical solution for the Fourier coeﬃcients X2n+1that is uni- formly valid with respect to the dimension of the circular holes. In fact, the norm of the completely continuous part His bounded by the estimate /bardblH/bardbl ≤max|τn|=τ0= 1−3 π<0.046/lessmuch1; this is uniformly valid in the parameter u0.The method of successive ap- proximations may be used to solve (3. 130); remarkably, only one step of the iteration process is needed to obtain an approximate analytical solution of high accuracy (3 to 4 correct digits in values of An). The result of one iteration is A2n+1/similarequal −2−1 2π1 2sinθ0Γ/parenleftbig n+1 2/parenrightbig Γ (n+ 1)P(1 2,0) n (cos 2θ0) ln [1 + sinθ0]−ln [cosθ0]. (3. 133) We may use (3. 133) to derive the magnetic ﬁeld distribution along the shell axis (z-axis). Due to symmetry we need only consider the positive z-axis (z≥0,θ= 0) and obtain Hm r(q,0) =−Φ πa2∞/summationdisplay n=0(2n+ 1)A2n+1q2n(3. 134) whereq=r/a. Use the tabulated value of the series [14] to rewrite (3. 134) in the form Hm r(q,0) =Φ πa2.π 2.sinθ0 ln [1 + sinθ0]−ln [cosθ0]× /parenleftbig 1 +q2/parenrightbig−3 22F1/parenleftBigg 3 4,5 4; 1;4q2cos2θ0 (1 +q2)2/parenrightBigg .(3. 135) ©200 1 CRC Press LLC Thehypergeometri cfunctio nin(3.135)admit saquadrati ctransformation totheLegendr efunction 2F1/parenleftBigg 3 4,5 4;1;4q2cos2θ0 (1+q2)2/parenrightBigg =/bracketleftbigg1+q2 R(q,θ0)/bracketrightbigg3 2 P1 2/bracketleftbigg1+q2 R(q,θ0)/bracketrightbigg (3.136) whereR(q,θ0)=/parenleftbig 1−2q2cos2θ0+q4/parenrightbig1 2;theLegendr efunctio nP1 2isrelated tothecomplet eellipti cintegra lofthesecon dkindEby(seeAppendix ,(B. 82)) P1 2/bracketleftbigg1+q2 R(q,θ0)/bracketrightbigg =2 πR0(q,θ0) R1 2(q,θ0)E/bracketleftbigg√4qcosθ0 R0(q,θ0)/bracketrightbigg , (3.137) whereR0(q,θ0)=/parenleftbig 1+2qcosθ0+q2/parenrightbig1 2. Itcaneasilybeshownthatifθ0/lessmuch1thevalueofthemagneti cﬁeldincreases inproportiontoθ−2 0.Represe ntativecalculation sofH−1 0H(q,0)areplotted inFigur e3.15.Computation sbaseduponFormulae(3.135)–(3 .137)andon the numerical solution of System (3. 130) were found to be in almost perfect agreement. In conclusion we remark that the growth of the magnetic ﬁeld concentra- tion at the apertures is restricted by some threshold value of the magneticﬁeld, the so-called critical value,H c.(This is characteristic for superconduc- tors of the ﬁrst kind, such as lead, tin, and niobium.) It is interesting thatthis phenomenon could be used for quite diﬀerent purposes, such as localisedconcentration of the magnetic ﬁeld, or attenuation (i.e., suppression) of themagnetic ﬁeld in some localised region of space. If the transition of the shell ( T >T c) to the superconducting state ( T≤Tc) is induced by a refrigeration process that starts from the equatorial zone ofthe shell, the initial frozen magnetic ﬂux is Φ e=πa2H0.As the supercon- ducting state occupies a larger part of the surface of the shell, the magnitudeof the magnetic ﬁeld increases, attaining its largest value on the apertureplanes where the refrigeration process terminates. By contrast, if the refrig-eration process starts at the shell rims, the initial frozen magnetic ﬂux is Φ r=πa2sin2θ0.H0, and the movement of the superconducting phase to the equatorial zone leads to the attenuation of the mean value of the magneticﬁeld because the frozen magnetic ﬂux has a constant value at any cross-section of the shell. 3.7 Screening number of superconducting shells In this section, we consider another example of a mixed boundary-value problem for Laplace’s equation in which Neumann boundary conditions are ©200 1 CRC Press LLC Figur e3.15 Frozenmagneti cﬁeldalongz-axis,forvariou sangle sθ0. speciﬁe donaspherica lshellsurface .Weconside rasuperconductin gshell, shapedasathinspherica lshellwithasinglecircula rhole.Itisplace dinan externa lmagnetostati chomogeneou sﬁeld→ H0,directe datangleαrelati veto thez-axis(seeFigur e3.16),whichistheaxisofrotationa lsymmetr yofthe shell. With no loss of generality, we may suppose that vector− →H0lies in a plane xOz, so that its vertical and horizontal components are H0 z=H0cosα≡H0 /bardbl, H0 x=H0sinα≡H0 ⊥. (3. 138) The magnetostatic potential function Ψ0(r,θ,ϕ ) describing this magnetic ﬁeld− →H 0=−∇Ψ0in spherical coordinates is Ψ0(r,θ,ϕ ) =−H0.r(cosαcosθ+ sinαsinθcosϕ) (3. 139) =−H0 /bardbl.rcosθ−H0 ⊥.rsinθcosϕ. (3. 140) ©200 1 CRC Press LLC0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1024681012141618 q = r / aH / H0100 200 600 Figure 3.16 Spherically-shaped superconducting shell. In the interior region 0 ≤r<a, the total potential has the form Ψ(i)=H0 /bardbl.a∞/summationdisplay n=0a(i) n/parenleftBigr a/parenrightBign Pn(cosθ) +H0 ⊥.a∞/summationdisplay n=1b(i) n/parenleftBigr a/parenrightBign P1 n(cosθ) cosϕ, (3. 141) whereas in the unbounded region r>a, the total potential has the form Ψ(e)= Ψ0+H0 /bardbl.a∞/summationdisplay n=0a(e) n/parenleftBigr a/parenrightBig−n−1 Pn(cosθ) +H0 ⊥.a∞/summationdisplay n=1b(e) n/parenleftBigr a/parenrightBig−n−1 P1 n(cosθ) cosϕ. (3. 142) As mentioned in the previous section, superconducting shells are modelled by ideal diamagnetic materials of zero relative permeability, so that the normal component of magnetic ﬁeld (in this case, Hr) vanishes at the shell surface. ©200 1 CRC Press LLC/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 αHH H0T00z θ0 IIo xy The continuity condition at r=atakes the form H(i) r(a,θ,ϕ ) =H(e) r(a,θ,ϕ ), θ∈(0,π), ϕ∈(0,2π), (3. 143) where the superscripts ianderefer to the interior and exterior regions, re- spectively. Furthermore, on the screen surface, the normal components satisfy H(i) r(a,θ,ϕ ) =H(e) r(a,θ,ϕ ) = 0, θ∈(θ0,π), ϕ∈(0,2π). (3. 144) Also we require continuity on the aperture ( r=a,θ∈(0,θ0), ϕ∈(0,2π)) for the other magnetic ﬁeld components: H(i) θ(a,θ,ϕ ) =H(e) θ(a,θ,ϕ ), (3. 145) H(i) ϕ(a,θ,ϕ ) =H(e) ϕ(a,θ,ϕ ). (3. 146) To these conditions are added the ﬁniteness of the energy integral /integraldisplay/integraldisplay/integraldisplay V/vextendsingle/vextendsingle/vextendsingle∇Ψ(i)/vextendsingle/vextendsingle/vextendsingle2 dV <∞, (3. 147) which determines the solution class for Fourier coeﬃcients a(i,e) nandb(i,e) n. Condition (3 .143) implies (for n= 1,2,3,...) na(i) n=−δ1n−(n+ 1)a(e) n, (3. 148) nb(i) n=−δ1n−(n+ 1)b(e) n. (3. 149) Enforcing the conditions (3 .144)−(3.146) leads to two independent systems of dual series equations for the internal Fourier coeﬃcients, ∞/summationdisplay n=12n+ 1 n+ 1a(i) nP1 n(cosθ) =−3 2sinθ, θ∈(0,θ0,) (3. 150) ∞/summationdisplay n=1na(i) nP1 n(cosθ) = 0, θ∈(θ0,π), (3. 151) and ∞/summationdisplay n=12n+ 1 n+ 1b(i) nP1 n(cosθ) =−3 2sinθ, θ∈(0,θ0), (3. 152) ∞/summationdisplay n=1nb(i) nP1 n(cosθ) = 0, θ∈(θ0,π), (3. 153) The ﬁnite energy condition (3 .147) requires ∞/summationdisplay n=1n 2n+ 1/vextendsingle/vextendsingle/vextendsinglea(i) n/vextendsingle/vextendsingle/vextendsingle2 <∞,∞/summationdisplay n=1n2(n+ 1) 2n+ 1/vextendsingle/vextendsingle/vextendsingleb(i) n/vextendsingle/vextendsingle/vextendsingle2 <∞, (3. 154) ©200 1 CRC Press LLC sothat/braceleftBig a(i) n/bracerightBig∞ n=1∈l2(0)and/braceleftBig b(i) n/bracerightBig∞n=0∈l2(2). TosolveEquation s(3.152)and(3.153),setxn=nb(i) nandintegrat e(3.153) usingFormula(B.49)(seeAppendix )toobtain ∞/summationdisplay n=12n+1 n(n+1)xnP1 n(cosθ)=−3 2sinθ,θ∈(0,θ0) (3.155) ∞/summationdisplay n=1xnPn(cosθ)=c1,θ∈(θ0,π) (3.156) wherec1istheconsta ntofintegration .FromtheDirichlet-Mehle rrepresen- tationforLegendr epolynomial s(1.149)wereadil ydeduc ereprese ntation sof thesametypeforassociatedLegendr efunctions: P1 n(cosθ)=2√ 2 π1 sinθn(n+1) 2n+1θ/integraldisplay 0sin(n+1 2)ϕsinϕ√cosϕ−cosθdϕ. (3.157) Now,followingthewell-establishe dprocedur edescri bedinSectio n2.1, transform (3 .155) and (3 .156) to the equations ∞/summationdisplay n=1xnsin/parenleftbigg n+1 2/parenrightbigg θ=/braceleftbigg−sin3 2θ, θ∈(0,θ0) c1sin1 2θ, θ∈(θ0,π). (3. 158) Exploit orthogonality of the trigonometric functions on (0 ,π) to obtain, for m= 1,2,..., xm=−R1m(θ0)−c1R0m(θ0) (3. 159) and, corresponding to m= 0,an equation for c1, 0 =−R10(θ0) + [1−R00(θ0)]c1, (3. 160) where Rnm(θ0) = 2 ˆQ(1 2,−1 2) n−1,m−1(cosθ0), (3. 161) with ˆQ(−1 2,1 2) nm denoting the usual normalised incomplete scalar product. Thus, the ﬁnal analytical form of the solution is xm=−/braceleftbigg R1m(θ0) +R10(θ0) 1−R00(θ0)R0m(θ0)/bracerightbigg . (3. 162) From (3. 161), it is evident that xm=O/parenleftbig m−1/parenrightbig asm→ ∞ ; henceb(i) m= O(m−2) asm→ ∞,and the obtained solution does in fact lie in l2(2). ©200 1 CRC Press LLC ThedualEquation s(3.150)and(3.151)fortheremainin gcoeﬃcie ntsa(i) n maybesolvedinvariou sways.Westartbyintegratin gbothequations: ∞/summationdisplay n=12n+1 n+1a(i) nPn(cosθ)=−3 2cosθ+c2,θ∈(0,θ0) (3.163) ∞/summationdisplay n=1a(i) n n+1P1 n(cosθ)=0,θ∈(θ0,π) (3.164) wherec2isanintegratio nconsta nttobedetermined .Indeducin g(3.164)we usedthewell-kn ownformula(seeAppendix ,(B.49)and(B.58)) Pn+1(x)−Pn−1(x) =−2n+ 1 n(n+ 1)/radicalbig 1−x2P1 n(x). Integrate Equation (3 .164) again to obtain ∞/summationdisplay n=1a(i) n n+ 1Pn(cosθ) =c3, θ∈(θ0,π) (3. 165) wherec3is another constant of integration to be determined. The dual series Equations (3 .163) and (3 .165) may be solved in various ways. We use a standard Abel integral transform to convert to equations with trigonometric kernels: ∞/summationdisplay n=12n+ 1 n+ 1a(i) ncos/parenleftbigg n+1 2/parenrightbigg θ=−3 2cos3 2θ+c2cosθ 2,θ∈(0,θ0) (3. 166) ∞/summationdisplay n=1a(i) n n+ 1sin/parenleftbigg n+1 2/parenrightbigg θ=c3sinθ 2, θ∈(θ0,π). (3. 167) The dual Equations (3 .166) and (3 .167) are equivalent to two systems of functional equations, ∞/summationdisplay n=12n+ 1 n+ 1a(i) ncos/parenleftbigg n+1 2/parenrightbigg θ=/braceleftbigg −3 2cos3 2θ+c2cos1 2θ, θ∈(0,θ0) c3cos1 2θ, θ ∈(θ0,π) (3. 168) and ∞/summationdisplay n=1a(i) n n+ 1sin/parenleftbigg n+1 2/parenrightbigg θ=/braceleftbigg −1 2sin3 2θ+c2sin1 2θ, θ∈(0,θ0) c3sin1 2θ, θ ∈(θ0,π).(3. 169) A retrospective justiﬁcation for the diﬀerentiation process in obtaining (3 .168) is needed, but none is needed for (3 .169). It is obvious that the solution of the ﬁrst equation lies in the required class ( l2), 2m+ 1 m+ 1a(i) m=−3 2Q1m(θ0) +c2Q0m(θ0)−c3Q0m(θ0), m≥1 (3. 170) ©200 1 CRC Press LLC whereQnm(θ0) =ˆQ(−1 2,1 2) nm (cosθ0) is the usual normalised incomplete scalar product. By considering the product of (3 .168) with cos1 2θ, the constants c2andc3 are related by −3 2Q10(θ0) +c2Q00(θ0) =c3[1−Q00(θ0)] . (3. 171) If the constants c2andc3are arbitrarily chosen, the solution of Equation (3.169) does not lie in the required class. The correct solution is found by requiring the function to be continuous at the point θ=θ0,leading to −1 2sin3 2θ0+c2sinθ0 2=c3sinθ0 2. (3. 172) From (3. 170), (3. 171), and (3. 172) we ﬁnally deduce a(i) m=−3 2m+ 1 2m+ 1/braceleftbigg Q1m(θ0)−sin3 2θ0 3 sin1 2θ0Q0m(θ0)/bracerightbigg . (3. 173) The closed form for the magnetostatic potential Ψ(i)(r,θ,ϕ ) is Ψ(i)(r,θ,ϕ ) =−3 2H0 /bardbl.a∞/summationdisplay n=1n+ 1 2n+ 1Q(1) 1n(θ0)/parenleftBigr a/parenrightBign Pn(cosθ) −H0 ⊥.acosϕ∞/summationdisplay n=11 nR(1) 1n(θ0)/parenleftBigr a/parenrightBign P1 n(cosθ) (3. 174) where R(1) 1n(θ0) =R1n(θ0) +R10(θ0) 1−R00(θ0)R0n(θ0), and Q(1) 1n(θ0) =Q1n(θ0)−sin3 2θ0 3 sin1 2θ0Q0n(θ0). A measure of screening eﬀectiveness of the superconducting open spherical shell is the screening number (recall that αdeﬁnes the direction of the external magnetic ﬁeld), K=H−1 0H(0,θ,ϕ) =/parenleftBig K2 /bardblcos2α+K2 ⊥sin2α/parenrightBig1 2, (3. 175) whereH(0,θ,ϕ) is the magnetic ﬁeld at the centre of the shell, and K/bardbl,K⊥ are screening numbers of the longitudinal and transverse magnetic ﬁeld, re- spectively. It is evident that K/bardbl=Q(1) 11(θ0), K⊥=R(1) 11(θ0). (3. 176) ©200 1 CRC Press LLC Figure 3.17 Longitudinal ( KL) and transversal ( KT) screening numbers for the spherically-shaped superconducting shell. Suppressing rather bulky details, the distribution of the magnetic ﬁeld, which penetrates into the screen, when taken along the axis of the screen (withq=r/a≤1) has components H(i) r/parenleftbigg q,/braceleftbigg 0 π/bracerightbigg ,ϕ/parenrightbigg =±H/bardblL(±q,θ0), H(i) θ/parenleftbigg q,/braceleftbigg 0 π/bracerightbigg ,ϕ/parenrightbigg =∓H⊥R(±q,θ0) cosϕ, H(i) ϕ/parenleftbigg q,/braceleftbigg 0 π/bracerightbigg ,ϕ/parenrightbigg =H⊥R(±q,θ0) sinϕ,(3. 177) ©200 1 CRC Press LLC0 10 20 30 40 50 60 70 80 90−140−120−100−80−60−40−200 θ0 , degreesSCREENING NUMBERS KL , KT , dBKT KL where 2πt2R(t,θ0) =−t 2sin2θ0−t−1arctan/bracketleftbiggtsinθ0 1−tcosθ0/bracketrightbigg +(1−t)3 sinθ0 1−2tcosθ0+t2+2t2arctan/bracketleftbiggsinθ0 t−cosθ0/bracketrightbigg +R10(θ0) 1−R00(θ0)t/braceleftbigg πR00(θ0)+(1−t)sinθ0 1−2tcosθ0+t2+arctan/bracketleftbiggsinθ0 t−cosθ0/bracketrightbigg/bracerightbigg , and 2π 3t2L(t,θ0) =−2 3tarctan/bracketleftbiggtsinθ0 1−tcosθ0/bracketrightbigg +/parenleftbig 1+t3/parenrightbig sinθ0 2(1−2tcosθ0+t2)+1+t 6sinθ0 +2 3t2arctan/bracketleftbiggsinθ0 cosθ0−t/bracketrightbigg −sin3 2θ0 3sin1 2θ0t(1+t) 2sinθ0 1−2tcosθ0+t2. Itfollowsfromth elastformul athatL(−1,θ0)=0;thi simplie sthat Hr(1,π,ϕ )=0,i.e.,th eboundaryconditio n(3.144)holdsatthispoint. Somecalculation susin gtheFor mula(3.176)areshowninFigur e3.17. These show that the transverse magnetic ﬁeld is less well shielded compared with the longitudinal magnetic ﬁeld. For instance, the shielding numbers ofa cavity with θ 0= 5◦have ratioK⊥/K/bardbl/revsimilar103(note the vertical scale is in decibels). ©200 1 CRC Press LLC Chapter4 ElectrostaticPotentialTheor yfor OpenSpheroidalShells Aftersphericalgeometry,spheroidalgeometrypr ovidesth esimples tsetting forthree-dimensionalpotentialtheory.Thischapterconsider sthepote ntial surroundingvariousopenspheroidalshel lstructures .Itpresentsasigniﬁcant extensionandgeneralisationofth esphericalshel lstudie sbecausevarious combinationsofcavitysizeandas pectratiooftheshellproduceextremely interestingstructuresforphysicalandengineeringapplications;th eholl ow cylinderison eexample. Astheratiobe tweenth eminorandmajoraxesincreases ,aclose dspheroidal surfacetakeswidelydiﬀeringshapesrangin gfromth ediskthroughtheoblate spheroid,tothesphere,throughth eprolatespheroid,toth elimitingformof athincylinde rofﬁnitelengt horofaneedle-shape dstructure. Whilstcuttingslotsinthespheroidalshellexpandsthepossibilitiesofmod- ellingofrealphysicalobjects ,itincrease stheanalyticalcomplexityofthe correspondin gboundary- valueproblem.Thi saccount sforthefactthat,un- tilnow,onlyth esimples tproblemsforconductorsdescribe dinspheroidal coordinateshavebeenanalyse dindetail ,namelyclose dspheroids(see,for example,[26])andspheroidalcap s[12]. Nevertheless,signiﬁcantprogresscanbemadeforaxiallysymmetricstruc- turesi nthissetting.Th eLaplaceoperatorseparatesi nthiscoordinatesystem, sothatdualortripleserie sequation scanbeconstructedbyenforcementof mixedboundaryconditionsontheconductingsurfac eortheapertureasap- propriate.Asexplainedi nChapter1,thes eequationsareequi valentto(and canbereformulate das)acertainﬁrst-kin dFredholmintegralequation.The originalﬁrst-kin dequationsmaybetransforme dtoaFredholmsecond-kind inﬁnitematri xequationbythemethodo fregularisation.Aswehavealready seen,th eregularisedsystemofequationspossesse smanydesirablefeatures includingrapidconvergenceofthesolution,obtaine dbytruncationmeth ods, totheexac tone,andguaranteedaccuracyofcomputations. Asfortheope nsphericalshellstudies,wewil lconside rspheroidalshellsin whichoneortwoaperture sareintroduce dinanaxisymmetricfashion.Prolate andoblatespheroidswithsu chaperture swillbediscussed.Afteranintro- ductoryformulation(Section4.1)ofmixedboundaryvalueproblemsinthe spheroidalcoordinatesystems ,weﬁrs texamin ethethin,perfectlyconduct- ing,prolatespheroidalshellwit honecircularhole(Section4.2).Th eprolate ©200 1 CRC Press LLC spheroidalshellinwhichalongitudinalslotisintr oducedtoproduceapair ofequallysize dspheroidalcapsisthenconsidere d(Section4.3) .Whenthe capsareoppositel ycharged,wemaycalculatethecapacitanceo ftheresulting condensor.Thecompleme ntarystructure,aprolatespheroidalshellwithtwo symmetricallydis posedcircularholes ,orspheroida lbarrel isdiscusse dinthe followingsection(4.4);thehollowrightcircularcylindermaybeviewe dasa limitingcase. Thenexttwosectionsexamin etheanalogousstructuresfor oblate spheroi- dalshellswit htwoapertures :theoblateshellwithalongitudinalslot,which produce sapairofequallysize dspheroidalcap s(Section4.5) ,andtheoblate spheroidalbarrel(Section4.6).Intheﬁnalsection ,thecapacitanceofth evar- iousshells(whe npositivelycharged )andcondensor s(comprisingop positely chargedcomponents)areexaminedasafunctionofas pectrati oandaperture size. Incontrasttoclosedstructures,thereh avebee nrelativelyfe wanalytical studiesofth eelectrostati cpote ntialdistributionsurroundingthreedimen- sionalo penstructureswithc avitiesan dedges .Viewedasanexampleofa three-dimensionalﬁniteopenconductorwithac avity,thes ecanonicalprob- lemsan dtheirsolutionsca nbeusedforthede velopmentan dtestin gofap- proximatemethod sofgeneralapplicabilityinpotentialcalculations. 4.1For mulatio nofmixe dboundar yvalueproblem sin spheroidalgeometry Asstate dintheIntr oduction,weconsiderinﬁnitelythin,perfectlycon- ducting,ope naxisymmetricspheroidalshell s(seeFigur e4.1)charge dtosome electrostaticpotential U.Weshal luseprolateandoblatespheroidalcoordi- natesinthetrigonometricc oordinatefor m(α,β,ϕ )describe dinSections1.1.4 and1.1.5.I nbot hcoordinatesystems,th esurfaceofeachshell S0lies on a coordinate surface α=const =α0(which is a spheroid), whilst the interval ofβdeﬁningS0depends on the particular structure. Thus, S0is deﬁned by α=α0, ϕ∈[0,2π],andβ∈I, whereIis a subinterval, or several disjoint subintervals of [0 ,π]; the comple- mentary interval I/prime= [0,π]\Iallows us to deﬁne the aperture or slot S1in the spheroidal surface by α=α0, ϕ∈[0,2π],andβ∈I/prime. Our aim is to construct the solution for electrostatic ﬁeld potential distribu- tionU(α,β,ϕ ) near the charged open shell S0when the potential is speciﬁed ©200 1 CRC Press LLC Figur e4.1 Spheroida lshellgeometry :prolat eandoblate intheformU(α0,β,ϕ)=f(β,ϕ)(forβ∈I)onthesurfac eoftheshell;we shallalsocalculat eitsassociatedcapacitanc eandsurfac echargedistribution. Thisboundar yvalueproble mofpotentialtheor yforspheroida lconductors maybeformulate dasdescri bedinSectio n1.3.Thus,weseekanelectrostatic potentialU(α,β,ϕ ) that is harmonic in R3, ∆U(α,β,ϕ ) = 0, (4. 1) which satisﬁes the Dirichlet boundary condition on the surface of the conduc- torS0, U(α0−0,β,ϕ ) =U(α0+ 0,β,ϕ ) =f(β,ϕ) forβ∈I,ϕ∈[0,2π],(4. 2) which has a normal derivative that is continuous across the slot S1, d dαU(α,β,ϕ )|α=α0+0 α=α0−0= 0 forβ∈I/prime,ϕ∈[0,2π], (4. 3) ©200 1 CRC Press LLC/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 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(4.4) Finally,th epotential Umusthaveboundedelectrostaticenergyinanyﬁnite volumeofspac eincludin gtheedgesofth econductor: W=1 2/integraldisplay/integraldisplay/integraldisplay V|gradU|2dV<∞. (4.5) AsnotedinSection1.3,anysolutionthatsatisﬁesal ltheseconditionsisnec- essarily unique and provides the physically relevant solution to this problem. In spheroidal coordinates, the method of separation of variables for La- place’s equation leads to partial solutions of the form (1. 31) or (1. 35) in prolate or oblate coordinates, respectively. We conﬁne attention to axisymmetric potential distributions (so∂ ∂ϕU= 0). Thus, the separation constant mof (1. 72) or (1. 74) is 0; furthermore the boundedness of the potential U(α,β) =U(α,β,φ ) requires that the separation constantnbe zero or a positive integer n= 0,1,2,.... Thus, the solution that satisﬁes Laplace’s equation, the continuity condi- tions on the boundary α=α0between the interior and exterior regions, and the decay condition at inﬁnity, takes the following form in prolate spheroidalcoordinates, U(α,β) = ∞/summationdisplay n=0C(p) nPn(cosβ)/braceleftbiggPn(coshα), 0≤α≤α0, Qn(coshα)Pn(coshα0)/Qn(coshα0), α>α 0,(4. 6) whilst in oblate spheroidal coordinates it takes the form U(α,β) = ∞/summationdisplay n=0C(o) nPn(cosβ)/braceleftbigg pn(isinhα), 0≤α≤α0, qn(isinhα)pn(isinhα0)/qn(isinhα0), α>α 0.(4. 7) Here,Pn(z),Qn(z) (z≥1) are the Legendre functions of the ﬁrst and second kind, respectively, Pn(cosβ) is a Legendre polynomial (with trigonometrical argument) and pn(z) =i−nPn(z),qn(z) =in+1Qn(z). The unknown (Fourier) coeﬃcients/braceleftBig C(p) n/bracerightBig∞ n=0and/braceleftBig C(o) n/bracerightBig∞n=0are to be found. Selecting the volume for integration Vin (4. 5) as the internal region of the spheroid ( α≤α0),the prolate geometry coeﬃcients must satisfy W=πd 2sinhα0∞/summationdisplay n=01 2n+ 1/vextendsingle/vextendsingle/vextendsingleC(p) n/vextendsingle/vextendsingle/vextendsingle2d dα[Pn(coshα)]2|α=α0<∞,(4. 8) ©200 1 CRC Press LLC wherea stheoblat egeometr ycoeﬃcie ntsmustsatisfy W=πd 2coshα0∞/summationdisplay n=01 2n+1/vextendsingle/vextendsingle/vextendsingleC(o) n/vextendsingle/vextendsingle/vextendsingle2d dα[pn(isinhα)]2|α=α0<∞.(4.9) Takingintoaccou nttheasymptoti cbehaviourofLegendr efunction sas n→∞(seeAppendix ,(B.72)and(B.73)),itfollowsfrom(4.8)and(4.9) thattherescale dcoeﬃcie nts A(p) n=C(p) nPn(coshα0),A(o) n=C(o) npn(isinhα0) (4.10) belongtothefunctiona lspaceofsquar esummabl esequence sl2: /braceleftBig A(p) n/bracerightBig∞ n=0,/braceleftBig A(o) n/bracerightBig∞n=0∈l2. (4.11) Thus,solution stothepotentialproble mwillbesoughtinthefollowing formforprolat espheroida lcoordinates, U(α,β)=∞/summationdisplay n=0A(p) nPn(cosβ)/braceleftbiggPn(coshα)/Pn(coshα0),α≤α0 Qn(coshα)/Qn(coshα0),α>α0/bracerightbigg (4.12) andforoblat espheroida lcoordinate sintheform U(α,β)=∞/summationdisplay n=0A(o) nPn(cosβ)/braceleftbiggpn(isinhα)/pn(isinhα0),α≤α0 qn(isinhα)/qn(isinhα0),α>α0/bracerightbigg .(4.13) Oncethecoeﬃcie ntsA(p) nandA(o) narefound ,theelectrostati cﬁeldpo- tentialU(α,β)isfullydetermine datanypointofthespace .Recal lthat axisymmetri cproblem sareconsidered .Therigorou ssolutio ntobedeveloped inthefollowingsection smakesitpossibl etoanalys eindetai lthepotential andelectrostati cﬁeldneartheconductor’ sedges. Thesurfac echargedensi tyσaccumulate dontheconducto rsurfac e(α= α0,β∈I)isdeﬁne dbythejumpinthenorma lcomponentEαoftheelectric ﬁeldacros sthesurfac e(cf.Equatio n(1.2)), σ(β)=1 4π{Eα(α0+0,β)−Eα(α0−0,β)}. (4.14) Thenorma lcomponentoftheelectri cﬁeld−→E=−gradUis Eα(α,β)|α=α0=h−1 αd dαU(α,β)|α=α0 (wher ehαisthemetri ccoeﬃcie nt),sousing(4.12),(4.13),andmetric coeﬃcie ntsinspheroida lcoordinate s(seeSectio n1.1.4),theexpressio nforσ in the prolate spheroidal system is σ(β) =1 4π1 d 2/radicalbig sinh2α0+ sin2β∞/summationdisplay n=0Λn(α0)A(p) nPn(cosβ), (4. 15) ©200 1 CRC Press LLC where Λn(α0) =/bracketleftbigg sinhα0/parenleftbiggQ/prime n(coshα0) Qn(coshα0)−P/prime n(coshα0) Pn(coshα0)/parenrightbigg/bracketrightbigg−1 . (4. 16) Employing the value of the Wronskian (B. 69) W(Pn,Qn)(z) =P/prime n(z)Qn(z)−Pn(z)Q/prime n(z) =/parenleftbig 1−z2/parenrightbig−1, we may simplify Λn(α0) = [sinhα0Pn(coshα0)Qn(coshα0)]−1. (4. 17) In the oblate spheroidal system, the charge density is σ(β) =1 4π1 d 2/radicalbig cosh2α0−sin2β∞/summationdisplay n=0λn(α0)A(o) nPn(cosβ), (4. 18) where the factor λn(α0) ={coshα0qn(isinhα0)pn(isinhα0)}−1(4. 19) arises from employing the value of the Wronskian of the pair pn,qn.It is worth noting that the surface charge density expressions (4. 15) and (4. 18) vanish for the range of βcorresponding to the aperture surface. The total charge Qon each isolated component of the conducting surface is obtained by integration of surface charge density σover the component surface. In considering particular problems, we will suppress the subscripts ( p) and (o) onAnwhen the context is unambiguous. In all calculations presented below, the semi-axial distance bis taken to be unity; thus, if the ratio a/bis speciﬁed, the interfocal distance dmay be determined. 4.2 The prolate spheroidal conductor with one hole Let us consider a prolate spheroidal shell S0with one circular hole deter- mined by an angle β0so thatS0is deﬁned by α=α0,0≤β≤β0,andφ∈[0,2π]. When charged to unit potential, enforcement of the mixed boundary condi- tions upon Equations (4. 12) determining the potential on the spheroidal shell ©200 1 CRC Press LLC produce sthedualserie sequations ∞/summationdisplay n=0AnPn(cosβ)=1,β ∈[0,β0], (4.20) ∞/summationdisplay n=0AnΛn(α0)Pn(cosβ)=0,β ∈(β0,π]. (4.21) Equation(4.20)describe sthepote ntialonS0,whereas(4.21)foll owsfrom thecontinuityofth enormalderivativeontheslot S1and Λn(α0)=/bracketleftbigg sinhα0/parenleftbiggQ/prime n(coshα0) Qn(coshα0)−P/prime n(coshα0) Pn(coshα0)/parenrightbigg/bracketrightbigg−1 . (4.22) Asnote dinSection4.1,thissimpliﬁesto Λn(α0)=[sinhα0Pn(coshα0)Qn(coshα0)]−1. (4.23) Letu sintroduceth eparameter εn=1−(2n+1)sin hα0Pn(coshα0)Qn(coshα0). (4.24) TheasymptoticsoftheLegendrefunction s(see(B.70)and(B.71))sh ow thatεnisasymptoticallysmall(as n→∞ ) εn=O(n−2)asn→∞. Deﬁneth enewcoeﬃcients xn=Λn(α0)An (2n+1)=An 1−εn, (4.25) sothat {xn}∞ n=0∈l2.Thesystem(4.20),(4.21)isthu sconvertedt othe standardform: ∞/summationdisplay n=0xn(1−εn)Pn(cosβ)=1,β ∈[0,β0], (4.26) ∞/summationdisplay n=0(2n+1)xnPn(cosβ)=0,β ∈[β0,π]. (4.27) Thissetofdualseriesequation shasalread ybee nconsideredinChapter1; itisaspecialcaseofthegeneralsetconsideredinSection2.1with α=β=0, m= 0, rn=εn, qn= 0, η=1 2.For these speciﬁc parameters, the Abel integral transform method essentially employs the Mehler-Dirichlet integrals, and the following pair of equations is obtained: ∞/summationdisplay n=0xn(1−εn) cos(n+1 2)β= cosβ 2, β∈[0,β0], (4. 28) ∞/summationdisplay n=0xncos(n+1 2)β= 0, β ∈[β0,π]. (4. 29) ©200 1 CRC Press LLC Wemayrewrit e(4.28)and(4.29)asaFourie rseriesexpressio nfora singlefunctio nFthatispiecewis edeﬁne dontwosubintervalsof[0,π], F(β)=∞/summationdisplay n=0xncos(n+1 2)β=/braceleftbigg F1(β)β∈[0,β0] 0,β∈[β0,π]/bracerightbigg , (4.30) where F1(β)=cos1 2β+∞/summationdisplay n=0xnεncos(n+1 2)β. Astandar dargume ntutilisin gcompletenes sandorthogonali typropertiesof thetrigonometri cfunction sproducesasecond-kin dsyste moflinearalgebraic equation sforthecoeﬃcie nts{xn}∞ n=0, xs−∞/summationdisplay n=0xnεnQns(β0)=Q0s(β0), (4.31) wheres=0,1,2,...,andQns(β0)≡ˆQ(−1 2,1 2) ns (cosβ0)istheusualnormalised incomplet escalarproduct. Thesyste m(4.31)hastheform (I−H)x=b whereHisacompletel ycontinuousoperato ronl2;thenormofHmaybe bounde duniforml ywithrespecttoβ0by /bardblH/bardbl≤max n|εn|=ε0=|1−sinhα0Q0(coshα0)|. (4.32) Considerin gthat Q0(coshα0)=1 2log/bracketleftbiggcoshα0+1 coshα0−1/bracketrightbigg >1 coshα0, (4.33) thenormisbounde dby N≤1−tanhα0<1. (4.34) Oneortwoiteration softhemeth odofsuccessi veapproximation sprovide anapproximat eanalytica lsolutio nthatismoreaccurat ewhenα0islarger, i.e.,thespheroi discloserinformtothesphere .Whentheeccentricitye issmall(e/lessmuch1,α0→∞)itispossibl etoshow,usingthehypergeometric represe ntation sofPn,Qn(seeAppendix ,(B.70)and(B.71)),that εn=−e2 2(2n−1)(2n+ 3)+O(e4), (4. 35) ©200 1 CRC Press LLC asn→∞.Accepting(4.35),thesolutionto(4.31)obtainedbythemethod ofsuccessiveapproximation sis xs=Q0s(β0)−e2 8Q0s(β0) (s−1 2)(s+3 2)− 1 2πe2 8/parenleftbigg 2sinβ0 2+2 3sin3 2β0/parenrightbigg/bracketleftbiggcos(s−1 2)β0 s−1 2−cos(s+3 2)β0 s+3 2/bracketrightbigg +O(e4).(4.36) Thecorres pondin gapproximationforth ecapaci tyC=bx0oftheopen chargedspheroidalconductoris C=b π(β0+sinβ0)+be2 24π/parenleftbigg 4β0+sinβ0−2sin2β0−1 3sin3β0/parenrightbigg +O(e4). (4.37) Theexpression(4.37)coincideswithth eresul t[12]obtainedbyadiﬀerent method .Itagreeswithth ecapacitanceofasphericalshellwhen e=0. Ifthevalueofth eeccentricity eisunrestricted,thesolutionto(4.31)is foundbytruncationtoaﬁnitesystemoflinearalgebraicequationsthatcan beeﬃcie ntlysolvednumerically.Fromamethodologicalpointofview,itis worthdemonstratinghowtoacceleratetheco nvergenc eofth esolutionofthe truncatedsystemtoth eexac tsolution.Theconvergenceratede pendsupon thebehaviou roftheparameter εn.Amor eprecisestateme ntofitsasymptotic behaviouris εn=−δ2/parenleftbigg n+1 2/parenrightbigg−2 +O(n−4),asn→∞, (4.38) whereδ2=(8sin hα0)−1.Withtheai mofmodifyin gtheSyste m(4.31),we introduceth enewparameter ε∗ n=εn+δ2(n+1 2)−2, (4.39) sothatε∗ n=O(n−4)asn→∞.Thetransformationtobeobtaine dismo- tivatedbytheobservationthat,ifoneneglects ε∗ n,theresultingdualseries equationscanbesolve dexactly.Asexplaine dinSection2.1,th edualequa- tions are then equivalent to a certain ordinary diﬀerential equation. Let g(β) =∞/summationdisplay n=0xn(n+1 2)−2cos(n+1 2)β, (4. 40) and f(β) =−cosβ 2−∞/summationdisplay n=0xnε∗ ncos(n+1 2)β. (4. 41) ©200 1 CRC Press LLC From Equation (4. 28) we deduce the second order diﬀerential equation g/prime/prime(β)−δ2g(β) =f(β), β ∈[0,β0]. (4. 42) Solving this equation (with g(0) =A,g/prime(0) = 0) produces the following ex- pression for g: g(β) =Acosh(δβ)−cosh(δβ)−cosβ 2 δ2+1 4−∞/summationdisplay n=0xnε∗ ncosh(δβ)−cos(n+1 2)β δ2+ (n+1 2)2, (4. 43) where A=∞/summationdisplay n=0xn (n+1 2)2. (4. 44) With the aid of these transformations, we may rewrite (4. 28) and (4. 29) in the ﬁnal form xm−∞/summationdisplay n=0xnε∗ nSnm(β0,δ) =S0m(β0,δ), (4. 45) wherem= 0,1,2,...,and Snm(β0,δ) =/braceleftbigg Qnm(β0) +τn(β0) γ(β0,δ)δ2Vm(β0)/bracerightbigg(n+1 2)2 (n+1 2)2+δ2, τn(β0) =1 (n+1 2)2/braceleftbigg cos(n+1 2)β0−(π−β0)(n+1 2) sin(n+1 2)β0/bracerightbigg , γ(β0,δ) = cosh(δβ0) + (π−β0)δsinh(δβ0), and Vm(β0) =2 π1 (m+1 2)2+δ2δcos(m+1 2)β0sinh(δβ0)+ 2 π1 (m+1 2)2+δ2(m+1 2) sin(m+1 2)β0cosh(δβ0). The truncation of the System (4. 45) is much more rapidly convergent than the truncation of the System (4. 31) because ε∗ ndecays more rapidly to zero than does εn.By determining the asymptotic behaviour of ε∗n,to O(n−6) terms, this procedure may be repeated to obtain another system with a further accelerated convergence rate; however, the complicated form of the system coeﬃcients hardly warrants the eﬀort since satisfactory solutions can be derived from the systems already obtained. We have computed the electrostatic ﬁeld distribution surrounding inﬁnitely thin prolate spheroidal conductors charged to unit potential by solving thesystem (4. 31) numerically (taking into account (4. 6) and (4. 25)). An ©200 1 CRC Press LLC Figur e4.2 Electrostati cpotentialnearaprolat espheroida lcap,charge dtounit potential,withparameter sa/b=0.2,β0=1300.Truncatio nnumber Ntr=11. exampl eisshowninFigur e4.2;theratioofmino rtomajoraxes,a/b= sinhα0/coshα0=0.2,andtheangula rsizeβ0oftheapertur eequal sto130o. Thetruncatio nnumberNtrwaschosentobe11. Computationall y,thesyste m(4.31)isveryattracti ve.Thesolutio nof thetruncate dsyste mconvergestotheexactsolutio n(thesolutio nofthe inﬁnit esystem )asNtr→∞.Theaccurac yofcalculation sundertruncation isillustrate dinFigur e4.3,wherenormalise derrorisplotte dasafunctio nof truncation number. The error is estimated in the maximum norm sense as e(Ntr) =max n≤Ntr/vextendsingle/vextendsinglexNtr+1 n−xNtrn/vextendsingle/vextendsingle max n≤Ntr/vextendsingle/vextendsingle/vextendsinglexNtrn/vextendsingle/vextendsingle/vextendsingle, where/braceleftbig xNtrn/bracerightbigNtr n=0denotes the solution to (4. 31) truncated to Ntrequations. A study of truncated solution accuracy conﬁrms that, in practice, for a wide ©200 1 CRC Press LLC−1.5 −1 −0.5 0 0.5 1 1.5−1.5−1−0.500.511.5 x/az/a 0.30.40.50.60.7 0.90.60.50.40.3 Figur e4.3 Normalise derrore(Ntr)asafunctio noftruncatio nnumberNtrfor theprolat espheroida lcap:(top)withaspectratioa/b=0.5and varyin gβ0;and(bottom )withβ0=130oandvaryin gaspectratio a/b. rangeofgeometrica lparameter sdescribin gtheconductor ,thetruncate dcoef- ﬁcientset{xn}Ntr n=0maybeobtaine dcorrectl ytothreedigits ,providedNtris approximatel yequalto10.Thisaccurac yissatisfactor yformostcalculations concernin gthepotential. Acorres pondingl yaccurat ecalculatio nofthesurfac echargedistribution require smoretermsthanforthepotential,asisevide ntbycomparin gEqua- tions(4.12)and(4.15),andtakin gintoaccou nttheasymptotic s(4.24)of thesmallparamete rεn.Sincetheseriesismuchlessrapidl yconvergentthan thatforthepotential,technique stoaccelerat etheconvergenc eoftheseriesare useful .Anexampl eofthesurfac echargedistributio nisshowninFigur e4.4 for the shell with ratio of minor to major axes, a/b= sinhα0/coshα0= 0.5, and the angular size of the aperture β0= 60◦. The truncation number Ntr was chosen to be 60, and the values were computed by a simple summation ©200 1 CRC Press LLC0 10 20 30 40 50 60 7010−1010−810−610−410−2 NtrTRUNCATION ERRORβ0= 300 β0= 800 β0= 1300 0 10 20 30 40 50 60 7010−1010−810−610−410−2 NtrTRUNCATION ERRORa / b = 0.1 a / b = 0.5a / b = 0.9 Figur e4.4 Surfac echargedensi tyσofaprolat espheroida lcap,charge dtounit potential,withparameter sa/b=0.5,β0=60o.Truncatio nnumber Ntr=60.Thedensi tywascompute dbysimpl esummatio nofthe Fourie rseries. ofthetruncate dFourie rseries ,sothatacontinuousapproximatio ntothe surfac echargeisobtained .Theoscillator yresult sareamanifestatio nofthe familia rGibbs ’phenomenon ;thesurfac echargeshoul dbezerooutsid ethe interval[−β0,β0].IfCes`arosummatio nisapplie d(see[9]),theoscillations aremuchsuppressed ,andoneobtain stheresult sofFigur e4.5.Excep tin the immediate vicinity of the edge a satisfactory representation of the surface charge is obtained. It is possible to improve the situation by estimating the leading order of the coeﬃcients in the inﬁnite system and exploiting a known inﬁnite sum whichrepresents the discontinuity exactly. In terms of the coeﬃcients x ndeﬁned in ©200 1 CRC Press LLC−150 −100 −50 0 50 100 150−0.100.10.20.30.40.50.60.7 θ0 , degreesSURFACE CHARGE DENSITY, σ Figure 4.5 Surface charge density σof a prolate spheroidal cap, charged to unit potential, with parameters a/b= 0.5,β0= 60o. Truncation number Ntr= 60. The density was computed by Ces` aro summation of the Fourier series. (4. 25), the surface charge is σ=1 4π1 d 2/radicalbig sinh2α0+ sin2β∞/summationdisplay n=0(2n+ 1)xnPn(cosβ), (4. 46) where the coeﬃcients xnsatisfy the System (4. 31); in accordance with (4. 27),σvanishes when β∈[β0,π].Upon writing Qsn(β0) =2 πcos/parenleftbig s+1 2/parenrightbig β0sin/parenleftbig n+1 2/parenrightbig β0 n+1 2+2 πs+1 2 n+1 2Rsn(β0),(4. 47) where Rsn(β0) =1 π/bracketleftbiggsin (s−n)β0 s−n−sin (s+n+ 1)β0 s+n+ 1/bracketrightbigg , ©200 1 CRC Press LLC−150 −100 −50 0 50 100 150−0.100.10.20.30.40.50.60.7 θ0 , degreesSURFACE CHARGE DENSITY, σ Figure 4.6 Surface charge density σof a prolate spheroidal cap, charged to unit potential, with parameters a/b= 0.5,β0= 60o. Truncation number Ntr= 11. The density was computed from Formula (4. 53). it is obvious that Qsn(β0) =2 πcos/parenleftbig s+1 2/parenrightbig β0sin/parenleftbig n+1 2/parenrightbig β0 n+1 2+O/parenleftbig n−2/parenrightbig (4. 48) asn→ ∞. Consider the system derived from (4. 31) by replacing Qsn(β0) with the leading term in (4. 48), i.e., neglecting the O/parenleftbig n−2/parenrightbig term: /tildewidexn=∞/summationdisplay s=0/tildewidexsεs2 πcos/parenleftbig s+1 2/parenrightbig β0sin/parenleftbig n+1 2/parenrightbig β0 n+1 2+2 πcos1 2β0sin/parenleftbig n+1 2/parenrightbig β0 n+1 2, (4. 49) wheren= 0,1,2,.... Its solution provides an asymptotic estimate for xnas n→ ∞ ; it may be established that xn−/tildewidexn=O/parenleftbig n−2/parenrightbig . ©200 1 CRC Press LLC−150 −100 −50 0 50 100 150−0.100.10.20.30.40.50.60.7 θ0 , degreesSURFACE CHARGE DENSITY, σ Thespecialformofthisasymptoti csystemallowsustodetermin eitssolution explicitly: /tildewidexn=2 πsin/parenleftbig n+1 2/parenrightbig β0 n+1 2D(α0,β0) (4.50) where D(α0,β0)=cos1 2β0+∞/summationdisplay s=0/tildewidexsεscos/parenleftbigg s+1 2/parenrightbigg β0 (4.51) isdetermine dbythesubstitutio nof(4.50)in(4.51). Rearrang ethesummatio nin(4.46)as ∞/summationdisplay n=0(2n+1)xnPn(cosβ0)= ∞/summationdisplay n=0(2n+1)/tildewidexnPn(cosβ0)+∞/summationdisplay n=0(2n+1)(xn−/tildewidexn)Pn(cosβ0).(4.52) Theﬁrsttermontheright-han dsideis ∞/summationdisplay n=0(2n+1)/tildewidexnPn(cosβ0)=4 πD(α0,β0)∞/summationdisplay n=0Pn(cosβ0)sin/parenleftbigg n+1 2/parenrightbigg β0 andmaybeevaluate dfromthewell-kn owndisco ntinuousseries ∞/summationdisplay n=0Pn(cosβ0)sin/parenleftbigg n+1 2/parenrightbigg β0=H(β0−β)/radicalbig 2(cosβ−cosβ0) derivedfromtheDirichlet-Mehle rFormula(1.124).(Hdenote stheHeaviside functio ndeﬁne dinAppendixA.)Thusthesurfa cechargeequals σ=1 4π1 d 2/radicalbig sinh2α0+sin2β× /braceleftBigg 2√ 2 πD(α0,β0)√cosβ−cosβ0H(β0−β)+∞/summationdisplay n=0(2n+1)(xn−/tildewidexn)Pn(cosβ)/bracerightBigg (4.53) Acalculatio nofthesurfa cechargedensi tyusing(4.53)isshowninFigure 4.6,usingthecoeﬃcie nts{xn}Ntr n=0obtaine dbysolvin gthesyste m(4.31) bythetruncatio nmeth odwithatruncatio nnumberNtrequal to 11. Two features are apparent. The current singularity at the edges is accurately represented; and the summation in (4. 53) has converged well. A sensitivetest of the accuracy of this result with 11 terms is the magnitude of thecalculated surface charge away from the conductor surface where the truesurface charge vanishes. The maximum error (or deviation from zero) in ©200 1 CRC Press LLC thisregioni slessthan0.5%ofth evalueatth etopofthecap.Ther eis novisibleimprovementtothegraphicalresultsas Ntrisincreased.Thus subtractionofanasymptoticallycorrectestimateofthesolutiontoth eSystem (4.31)providesamuchmorerapidlyconvergentserie sthanth eﬁrstestimate obtainedsimplybytruncation ;thisobservationals oremainstru eiftheﬁrst estimateisreplacedbyanestimateobtainedbyCes`arosummation. 4.3Theprolat espheroidalconductorwithalongitudinal slot Inthissectionweconsideraprolatespheroidalsurfac einwhichalongitu- dinalslothasbee ncut,toproducetwospheroidalcapsofequalsize;theyare speciﬁedby α=α0,β∈(0,β0)∪(π−β0,π),φ∈(0,2π). Thegeometryissh owninFigur e4.1b.Assum ethatthesetwosegme ntsare chargedtoconsta ntpotentials U1andU2,respecti vely. Enforcementon(4.13)ofth eboundar yconditions U(α0−0,β)=U(α0+0,β)=U1,forβ∈[0,β0], (4.54) U(α0−0,β)=U(α0+0,β)=U2,forβ∈[π−β0,π], (4.55) andofthecontinuityofthenormalderi vativeofth epote ntialonth eslot, d dαU(α,β)|α=α0+0 α=α0−0=0,forβ∈(β0,π−β0), (4.56) leadstothefollowin gsymmetrictripleserie sequation swithLegendr epoly- nomialkernels, ∞/summationdisplay n=0AnPn(t)=U1,t∈(t0,1], ∞/summationdisplay n=0Λn(α0)AnPn(t)=0,t∈(−t0,t0), (4.57) ∞/summationdisplay n=0AnPn(t)=U2,t∈[−1,−t0), wheret=cosβ,t0=cosβ0.Thesystem(4.57)i sparticularcas eofthe equationsofTypeAdescri bedi nSection2.4.1(Legendrepolynomial sare Jacobi polynomials P(α,α) n withα= 0),so the method described may be exploited to solve (4. 57). ©200 1 CRC Press LLC Wenowconsidertwoparticularcases, U1=U2=1an dU1=−U2=1. Obviously,caseswithanyotherconstantvaluesofthepotentials U1andU2 canbededucedfromthes esolutions .Fromapracticalpointofview,when U1=U2,thetwopartsoftheprolatespheroidalconductorwithalongitudinal slotmusttoconnecte dbyathinwireinordertoallowchargingt oequal potential;howe ver,wemayassum ethatthi swireissothinthattheinﬂuence ofitselectricﬁeldcanbeneglected.When U1=−U2thisstructuremodelsa condensororcapacitorwithplatesintheformofspheroidalcaps. Thesymmetr ypropertyofLegendr epolynomials, Pn(−t)=(−1)nPn(t), maybeappliedtoestablishtwodecoupledsystemsofdualseriesequations fortheeve n(l=0)an dodd(l=1)inde xcoeﬃcients,res pectively,deﬁned on[−1,0]: ∞/summationdisplay n=0A2n+lP2n+l(t)=(−1)l,t∈[−1,−t0), ∞/summationdisplay n=0Λ2n+l(α0)A2n+lP2n+l(t)=0,t ∈(−t0,0). (4.58) Therelation(2.131)connectsJacob ipolynomial sandLegendrepolynomi- als, P2n+l(t)=tlP(0,l−1 2) n (2t2−1), sosettin gu=2t2−1,u0=2t2 0−1wem aytransform(4.58)todualseries equationsdeﬁnedove rthecompleterange[ −1,1]ofth enewvariabl e: ∞/summationdisplay n=0Λ2n+l(α0)A2n+lP(0,l−1 2) n (u)=0,u∈(−1,u0), (4.59) ∞/summationdisplay n=0A2n+lP(0,l−1 2) n (u)=(−1)l/parenleftbigg1+u 2/parenrightbigg−l 2 ,u∈(u0,1). (4.60) ThedualseriesEquation s(4.59)and(4.60)wereconsidere dinSection2.1. Omitting some details let us illustrate the main stages of the argument in this particular case. The Abel integral representations for the Jacobi polynomials(1. 171)–(1. 174) are /integraldisplay u −1(1 +t)l−1 2P(0,l−1 2) n (t)dt=Γ(n+l+1 2)√πΓ(n+l+ 1)/integraldisplayu −1(1 +x)lP(−1 2,l) n (x)dx (u−x)1 2 (4. 61) and P(0,l−1 2) n (u) =Γ(n+ 1)√πΓ(n+1 2)/integraldisplay1 u(1−x)−1 2P(−1 2,l) n (x)dx (x−u)1 2. (4. 62) ©200 1 CRC Press LLC The functional equations are then converted to the following form: ∞/summationdisplay n=0Λ2n+l(α0)A2n+lΓ(n+l+1 2) Γ(n+l+ 1)P(−1 2,l) n (u) = 0, u∈(−1,u0),(4. 63) ∞/summationdisplay n=0A2n+lΓ(n+ 1) Γ(n+1 2)P(−1 2,l) n (u) =(−1)l √π/parenleftbigg1 +u 2/parenrightbigg−l , u∈(u0,1).(4. 64) A suitable small parameter may now be identiﬁed in the Equation (4. 63) as ε2n+l= 1−Λ2n+l(α0) 4Γ(n+1 2)Γ(n+l+1 2) Γ(n+ 1)Γ(n+l+ 1). (4. 65) It is asymptotically small: ε2n+l=O(n−2) asn→ ∞.The unknowns are rescaled according to x2n+l=A2n+lΓ(n+ 1) Γ(n+1 2)/braceleftBig h(−1 2,l) n/bracerightBig1 2, (4. 66) where/braceleftBig h(−1 2,l) n/bracerightBig1 2is the norm of the Jacobi polynomials; thus {x2n+l}∞ n=0∈ l2. Equations (4. 63) and (4. 64) may now be written in the form F(u) =∞/summationdisplay n=0x2n+lˆP(−1 2,l) n (u) =/braceleftbiggF1(u), u∈(−1,u0) F2(u), u ∈(u0,1)/bracerightbigg , (4. 67) where F1(u) =∞/summationdisplay n=0x2n+lε2n+lˆP(−1 2,l) n (u), F2(u) = (−1)lπ−1 22l(1 +u)−l. Exploiting orthogonality of the normalized Jacobi polynomials ˆP(−1 2,l) n leads, as usual, to the second-kind inﬁnite system of linear algebraic equations for the unknowns {x2n+l}∞ n=0, (1−ε2m+l)x2m+l+∞/summationdisplay n=0x2n+lε2n+lˆQ(−1 2,l) nm (u0) =/braceleftBigg 23 4π−1 2ˆQ(−1 2,0) 0m(u0), ifl= 0 −2π−1 2/braceleftbig (m+ 1)(m+1 2)/bracerightbig−1 2√1−u0ˆP(1 2,0) m(u0),ifl= 1(4. 68) wherem= 0,1,2,..., and ˆQ(−1 2,l) nm (u0) is the incomplete scalar product of normalised Jacobi polynomials. ©200 1 CRC Press LLC Figur e4.7 Electrostati cpotentialnearaslotte dprolat espheroida lshell,both componentscharge dtounitpotential.Thegeometrica lparameters area/b=0.5,β0=60o.Truncatio nnumberNtr=11. Becaus ethematri xoperato rofthesyste m(4.68)isacompletel ycontinu- ousperturbatio noftheidentity,thesequenc e{x2n+l}∞ n=0israpidl yconvergent andthetruncatio nmeth odisveryeﬃcie ntinsolvin gthissyste mnumericall y. Thebehaviourofthenormalise derrorasafunctio noftruncatio nnumber isverysimila rtothatconsidere dinthepreviou ssectio n(seeFigure s4.3); typicall y,Ntr=10equation ssuﬃc etoproducecoeﬃcie ntsolution swith3 correc tdigitsforawiderangeofaspectratios(inde pendentofapertur esize). Asanillustratio nofthenumerica lprocess,thedistributio nofelectrostatic ﬁeldpotentialnearthespheroida lconducto rwithalongitudina lslotcharged tounitpotential(U1=U2=1,l=0in(4.68))isshowninFigur e4.7;the ratio of minor to major axes, a/b= sinhα0/coshα0= 0.5 and the angular size of each cap is β0= 60◦; and the system truncation number Ntrwas taken to be 11. The potential near the spheroidal condensor in which the upper and lower ©200 1 CRC Press LLC−1.5 −1 −0.5 0 0.5 1 1.5−1.5−1−0.500.511.5 x/az/a 0.4 0.50.6 0.70.8 0.90.80.9 0.70.60.5 0.4 Figur e4.8 Electrostati cpotentialneartheprolat espheroida lcondenser ,the plate scharge dtounitpositiveandnegati vepotential.Thegeomet- ricalparameter sarea/b=0.5,β0=60o.Truncatio nnumberNtr=11. plate sarecharge dtopotentialsU1=1andU2=−1(sol=1in(4.68))is displayedinFigur e4.8;thegeometrica lparameter sarea/b=0.5,β0= 60◦, and a truncation number Ntr= 11 was used. Whenβ0=π 2(u0=−1) the aperture in the conductor closes, becoming a closed spheroidal shell charged to unit potential ( l= 0), the system (4. 68) has the explicit solution x2m= (−1)m23 4√πˆQ(−1 2,0) 0m(−1) = 0 (m> 0), x 0=23 4√π, (4. 69) from which follows the representation of the electrostatic potential in closed form: U(α,β) =Q0(coshα) Q0(coshα0)forα≥α0,β∈[0,π]. (4. 70) It is readily veriﬁed that this is indeed the correct potential. ©200 1 CRC Press LLC−1.5 −1 −0.5 0 0.5 1 1.5−1.5−1−0.500.511.5 x/az/a −0.2 −0.4−0.6 −0.80.20.40.60.8 Letusconsid erthetransitio nfromspheroi dtospher eofradiu sa.Spheroi- dalcoordinate s(α,β,ϕ)degenerat etospherica lcoordinat es(r,θ,ϕsp)ifthe identiﬁcations θ=β,ϕsp≡ϕ,r=1 2d 2eα,a=1 2d 2eα0 aremadeinsuchawaythatasd 2→0,α→∞,andα0→∞,theproducts remai nﬁnite .Itmaybecheckedthatthesolutio nreduc estothatforthe spherica lconducto r(analys edinSectio n3.2).Infact,thelimits lim α0→∞sinhα0Q2n+l(coshα0)P2n+l(coshα0)=(4n+2l+1)−1, lim α0→∞sinhα0Q0(coshα0)=1, (4.71) arevalid(seeAppendix ,(B.70)and(B.71)),soacompariso nof(4.68)with thesimila rsyste mintheSectio n3.2showstheidentityofthesolutions .In calculatin gtheelectrostati cﬁelditshoul dbenotedthatasα,α0→∞,the followingreplaceme ntaremade: P2n+l(coshα) P2n+l(coshα0)→/parenleftBigr a/parenrightBig2n+l ,Q2n+l(coshα) Q2n+l(coshα0)→/parenleftBigr a/parenrightBig−2n−l−1 . (4.72) Thelimitin greprese ntation s(4.71)and(4.72)followfromtheasymptotic behaviouroftheLegendr efunction s(whenα,α0→∞,see[1]). 4.4Theprolat espheroida lconducto rwithtwocircular holes Inthissectio nweconside rthecompleme ntarystructur etotheslotted spheroi dofthepreviou ssection ,andsupposethatthespheroida lconductor hastwocircula rholes(seeFigur e4.1(c)).TheshellS0is deﬁned by α=α0,β∈(β0,π−β0),φ∈(0,2π) ; whena/b/lessmuch1,it may be visualised as a spheroidal cylinder. It is charged to unit potential, so U(α0−0,β) =U(α0+ 0,β) = 1,β∈[β0,π−β0], (4. 73) whereas the normal derivative of the potential is continuous on the apertures, d dαU(α,β)|α=α0+0 α=α0−0= 0,β∈(0,β0)∪(π−β0,π). (4. 74) Enforcing the boundary conditions (4. 73) and (4. 74) on (4. 12) produces a set of symmetric triple series equations of Type B (2. 126)–(2. 128) from which ©200 1 CRC Press LLC maybededuced(inthesam ewayasforEquations(4.58))thedualseries equationsdeﬁnedove rthehalfrange[ −1,0](settingt=cosβ,t0=cosβ0): ∞/summationdisplay n=0Λ2n(α0)A2nP2n(t)=0,t∈(−1,−t0), ∞/summationdisplay n=0A2nP2n(t)=1,t∈(−t0,0). (4.75) Followingth esam eargumentasinSection4.3,wemayreduc etheEquations (4.75)t odualserie sequationsinvolvingJacob ipolynomial sdeﬁnedover theinterval[ −1,1].Setting u=2t2−1,u0=2t2 0−1,andβ1=π 2−β0, u1=cos2β1=−u0,weobtain ∞/summationdisplay n=0(−1)nΛ2n(α0)A2nP(−1 2,0) 2n (u)=0,u∈(−1,u1), (4.76) ∞/summationdisplay n=0(−1)nA2nP(−1 2,0) 2n (u)=1,u∈(u1,1). (4.77) ThegeneraltreatmentexpoundedinSection2.4.2ofdualequationsofthis type,arisingfromTypeBtripleseries,di dnotcove rthepair(4.76),(4. 77).Letuss peciﬁcallydemonstrateh owtotreatthes eequations.Before employin gtheintegralrepresentationsofAbel’stypeforJacobipolynomials, integratetheEquation(4.76)withtheweight(1 −u)−1 2,usin gthevariant (2.36)ofRodrigues ’formula.(Althoughthi sintegrationcomplicate sthe solutionprocess,itisabsolutel ynecessarybecauseadirec tapplicationofthe integralrepresentationsofAbe ltypewouldresulti ntheoccurrenceofthe Jacobipolynomialkernels P(−1,1 2) n forwhichthetheorydevelopedinSection 2.1isnotvalid.) The transform method may now be applied in a standard manner, similar to that in the previous section, to obtain the expansion of some function F in a Fourier series over the complete orthogonal system of Jacobi polynomials/braceleftBig ˆP(0,3 2) n/bracerightBig∞ n=1, piecewise deﬁned over two subintervals of [ −1,1] : F(u) =∞/summationdisplay n=1x2nˆP(0,3 2) n−1(u) =/braceleftbigg F1(u), u ∈(u1,1) F2(u), u∈(−1,u1)/bracerightbigg , (4. 78) where F1(u) = 2√ 2π(1−A0)(1 +u)−3 2+∞/summationdisplay n=1x2nε2nˆP(0,3 2) n−1(u), F2(u) =−A0Λ0(α0)√π/braceleftBigg 2 (1 +u)+√ 2 (1 +u)3 2ln/bracketleftBigg√ 2−√1 +u√ 2 +√1 +u/bracketrightBigg/bracerightBigg . ©200 1 CRC Press LLC Here x2n=(−1)n 4A2nΛ2n(α0)Γ(n+1) Γ(n+3 2)h(0,3 2) n−1/bracketleftBig h(1,1 2) n−1/bracketrightBig−1 2, (4.79) andtheasymptoticallysmallparamete ris ε2n=1−4 Λ2n(α0)n(n+1 2)/bracketleftbiggΓ(n+1 2) Γ(n+1)/bracketrightbigg2 =O(n−2)asn→∞.(4.80) Theconstant A0isdeterminedbyenforcin gcontinui tyonF(u)atu1, A0=1 g(u1)/bracketleftBigg 1+∞/summationdisplay n=1x2nε2nQn(u1)/bracketrightBigg , (4.81) where g(u1)=1−1 πsinhα0Q0(coshα0)/braceleftBigg/radicalbigg 1+u1 2+1 2log/bracketleftBigg√ 2−/radicalbig (1+u1)√ 2+/radicalbig (1+u1)/bracketrightBigg/bracerightBigg and Qn(u1)=1√π/bracketleftbigg1 2(1+u1)/bracketrightbigg3 2/bracketleftbigg n(n+1 2)/bracketrightbigg−1 2ˆP(0,3 2) n−1(u1). TheEquation(4.78)isnowtransformedinthesamewayas(4.67),taking intoaccou nt(4.81).Theﬁnalformofthei.s.l.a.e.is x2m+∞/summationdisplay n=1x2nε2n/braceleftBigg ˆQ(1,1 2) n−1,m−1(u1)−2√ 2Qn(u1)Qm(u1) g(u1)sinhα0Q0(coshα0)/bracerightBigg =2√ 2Qm(u1) g(u1)sinhα0Q0(coshα0),(4.82) wherem=1,2,....Th esyste m(4.82)possesse sthesam efeature sasthe system(4.68).Thenormofth ecompletel ycontinuouspart Hofthematrix operatorin(4 .82)isuniforml ybounde d(withrespec ttoth eparameters)by theestimate /bardblH/bardbl≤max n|εn|=ε1. Thei.s.l.a.e.(4.82)iseﬀectivelysolvednumericallybythetruncation method .Thebehaviourofsolutionaccurac yasafunctionoftruncationnum- berisver ysimilartothatdescribe dintheprevioussections.Computedresults ofthepote ntialdistributionnearth eprolatespheroidalconductorwithtwo holeswhenchargedtotheuni tpote ntialareshowni nFigur e4.9.Thegeo- metrical parameters are a/b= 0.5,β0= 30◦; a truncation number Ntr= 11 was used. ©200 1 CRC Press LLC Figure 4.9 Electrostatic potential near a prolate spheroidal barrel charged to unit potential with geometrical parameters a/b= 0.5,β0= 30o.Trun- cation number Ntr= 11. Ifβ0= 0 (so that β1=π 2,u1=−1), the limiting case of a closed spheroidal shell is obtained; from (4. 81) and (4. 82) we see that x2m= 0 (m= 1,2,...), A 0= 1. Thus, the electrostatic potential near the closed spheroidal shell has the form U(α,β) =Q0(coshα) Q0(coshα0)forα≥α0,β∈[0,π]. (4. 83) This expression (4. 83) agrees with the expression (4. 70) that was obtained for the limiting case of the spheroidal shell with a closing narrow slot. ©200 1 CRC Press LLC−1.5 −1 −0.5 0 0.5 1 1.5−1.5−1−0.500.511.5 x/az/a 0.50.60.70.8 0.90.90.8 0.70.6 0.5 4.5Theoblatespheroidalconductorwithalongitudinal slot Inthissectionweconsideranoblatespheroidalsurfaceinwhichalongitu- dinalslothasbeencu ttoproducetwospheroidalcapsofequalsize ;theyare speciﬁedby α=α0,β∈(0,β0)∪(π−β0,π),φ∈(0,2π) TheshellS0(seeFigure4.1(d ))isth eoblateanalogueofthestructureconsid- eredinSection4.3,andcomprisestwosymmetricaloblatespheroidalsegme nts thatar eassumedtobechargedtotheconstantpote ntialvalues U1=1and U2=(−1)l(l=0,1).Thenthemixe dboundar ycondition s(similarto(4. 54)–(4.56))taketheform U(α0−0,β)=U(α0+0,β)=1,β∈[0,β0], (4.84) U(α0−0,β)=U(α0+0,β)=(−1)l,β∈[π−β0,π], (4.85) d dαU(α,β)|α=α0+0 α=α0−0=0,β∈(β0,π−β0). (4.86) Enforcingtheseboundar ycondition son(4.13)pr oducesthefoll owing functionalequation son[−1,0]: ∞/summationdisplay n=0A2n+lP2n+l(t)=(−1)l,t∈[−1,−t0), (4.87) ∞/summationdisplay n=0λ2n+l(α0)A2n+lP2n+l(t)=0,t∈(−t0,0)(4.88) where,asnote dinSection4.1,th efactor λn(α0) ={coshα0qn(isinhα0)pn(isinhα0)}−1(4. 89) arises from employing the value of the Wronskian of the pair pn,qn.This system is identical to the prolate spheroidal shell system (4. 58) except for the replacement of the factor Λ2n+l(α0) ={sinhα0Q2n+l(coshα0)P2n+l(coshα0)}−1 by λ2n+l(α0) ={coshα0q2n+l(isinhα0)p2n+l(isinhα0)}−1(4. 90) in (4. 88). With this replacement, the solution of the dual series Equations(4. 87)–(4. 88) is identical to that obtained in the prolate case yielding the ©200 1 CRC Press LLC i.s.l.a.e. (1−ε2m+l)x2m+l+∞/summationdisplay n=0x2n+lε2n+lˆQ(−1 2,l) nm(u0) =/braceleftBigg 23 4π−1 2ˆQ(−1 2,0) 0m(u0), ifl=0, −2π−1 2/braceleftbig (m+1)(m+1 2)/bracerightbig−1 2√1−u0ˆP(1 2,0) m(u0),ifl=1,(4.91) wherem=0,1,2,..., ε2n+l=1−λ2n+l(α0) 4Γ(n+1 2)Γ(n+l+1 2) Γ(n+1)Γ(n+l+1)=O(n−2)asn→∞,(4.92) andalltheotherdeﬁnition sandrelation sarethesameasin(4.68).The validityoftheasymptoti cestimat e(4.92)isestablishe dbythebehaviour ofthefunction sqn(isinhα0),pn(isinhα0)asn→∞(seeAppendix ,(B.70) and(B.71)). Switchingtothecompleme ntaryangleβ1=π 2−β0,withu1=cos2β1= −u0,wesety2n+l=(−1)nx2n+l,anduse(B.170)toobtai nanothe rconve- nientformofthesystem, y2m+l−∞/summationdisplay n=0y2n+lε2n+lˆQ(l,−1 2) nm(u1) =  23 4π−1 2/bracketleftBig δ0m−ˆQ(0,−1 2) 0m(u1)/bracketrightBig , ifl=0, −2π−1 2√1+u1/bracketleftbig (m+1)(m+1 2)/bracketrightbig−1 2ˆP(0,1 2) m(u1),ifl=1.(4.93) Asanillustratio nofthenumerica lprocess,thespatia ldistributio nofelec- trostati cﬁeldpotentialnearthelongitudinall yslotte dconductor ,withboth componentscharge dtounitpotential(U1=U2=1,l=0in(4.93)),isshown inFigur e4.10;theratioofmajortomino raxesisa/b=coshα0/sinhα0=2.0, andtheangula rsizeofeachcomponentisβ0=60◦;thesyste mtruncation numberNtrwaschosentobe11.Asafunctio noftruncatio nnumber,the accurac yofsolution stothesyste m(4.93)aftertruncatio nhasthesame genera lbehaviourasdescri bedfortheprolat espheroida lshellsconsidere din earlie rsections. Whenthecomponentsareoppositel ycharged ,thestructur eactsasacon- densor .Thepotentialneartheslotte doblat espheroida lshell,inwhichthe upperandlowerplate sarecharge dtopotentialsU1=1andU2=−1(so l=1in(4.93)),isdisplayedinFigur e4.11;thegeometrica lparameter sare a/b= 2 andβ0= 60◦,and a truncation number Ntr= 11 was used. As expected, the electrostatic ﬁeld is strongly conﬁned to the interior. The closed oblate spheroidal shell ( β0=π 2,l= 0), charged to unit potential, has the explicit solution obtained from (4. 93): y2m=1√π23 4δ0m, m≥0, ©200 1 CRC Press LLC Figure 4.10 Electrostatic potential near a slotted oblate spheroidal shell, both components charged to unit potential. The geometrical parameters area/b= 2,β0= 60o.Truncation number Ntr= 11. so the closed form of the potential distribution is U(α,β) =q0(isinhα) q0(isinhα0),forα≥α0,β∈[0,π]. This is in accord with the known solution [26]. Let us consider the transition from oblate spheroid to sphere of radius a. In a similar way to that discussed for the prolate case, oblate spheroi- dal coordinates ( α,β,ϕ ) degenerate to spherical coordinates ( r,θ,ϕ sp) if the identiﬁcations θ=β,ϕ sp≡ϕ,r=1 2d 2eα,a=1 2d 2eα0 are made in such a way that, asd 2→0,α→ ∞ andα0→ ∞,the products remain ﬁnite. It may be checked that the same solution as obtained for the ©200 1 CRC Press LLC−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−2.5−2−1.5−1−0.500.511.522.5 x/az/a 0.60.70.80.90.96 0.960.90.80.70.6 0.5 Figur e4.11 Electrostati cpotentialneartheoblat espheroida lcondenser ,the plate scharge dtounitpositiveandnegati vepotential.Thegeomet- ricalparameter sa/b=2,β0=60o.Truncatio nnumberNtr=11. spherica lconducto r(Sectio n3.2)isfound .Infact,thelimits lim α0→∞coshα0q2n+l(isinhα0)p2n+l(isinhα0)=(4n+2l+1)−1, lim α0→∞coshα0q0(isinhα0)=1, (4.94) arevalid(seeAppendix ,(B.70)and(B.71)),soacompariso nof(4.93) withthesimila rsyste minSectio n3.2showstheidentityofthesolutions .In calculating the electrostatic ﬁeld it should be noted that as α,α 0→ ∞ , the following replacements are made: p2n+l(isinhα) p2n+l(isinhα0)→/parenleftBigr a/parenrightBig2n+l ,q2n+l(isinhα) q2n+l(isinhα0)→/parenleftBigr a/parenrightBig−2n−l−1 .(4. 95) The limiting representations (4. 94) and (4. 95) follow from the asymptotic behaviour of the Legendre functions (when α,α 0→ ∞,see [1]). ©200 1 CRC Press LLC−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−2.5−2−1.5−1−0.500.511.522.5 x/az/a −0.4−0.6−0.8−0.20.20.40.6 0.8 0.2 4.6Theoblatespheroida lconductorwit htwocircular holes Inthissectionweconsiderthestructur ecomplementarytoth eslotte doblate spheroidofthepreviou ssection .Thegeometryofanoblatespheroidalshell withtwoequalcircularholesi sshowninFigur e4.1(e).Theshell S0is deﬁned by α=α0,β∈(β0,π−β0),φ∈(0,2π) ; and is assumed to be charged to unit potential. The mixed boundary condi- tions are U(α0−0,β) =U(α0+ 0,β) = 1,β∈(β0,π−β0) (4. 96) andd dαU(α,β)|α=α0+0 α=α0−0= 0,β∈(0,β0)∪(π−β0,π). (4. 97) Enforcement of the boundary conditions (4. 96) and (4. 97) on (4. 13) produces symmetric triple series equations; a standard argument reduces these to the following dual series equations deﬁned over [ −1,0], wheret= cosβ, t0= cosβ0: ∞/summationdisplay n=0λ2n(α0)A2nP2n(t) = 0,t∈(−1,−t0), (4. 98) ∞/summationdisplay n=0A2nP2n(t) = 1,t∈(−t0,0) (4. 99) where the factor λ2n(α0) ={coshα0q2n(isinhα0)p2n(isinhα0)}−1(4. 100) arises from employing the value of the Wronskian of the pair pn,qn. This system is identical to the prolate spheroidal shell system (4. 75), except for the replacement of the factor Λ 2n(α0) byλ2n(α0) in (4. 98). With this replacement, the solution of the dual series Equations (4. 98) and (4. 99)is identical to that obtained in the prolate case. Thus, mutatis mutandis, we obtain the i.s.l.a.e. x 2m+∞/summationdisplay n=1x2nε2n/braceleftBigg ˆQ(1,1 2) n−1,m−1(u1)−2√ 2Qn(u1)Qm(u1) g(u1) coshα0q0(isinhα0)/bracerightBigg =2√ 2Qm(u1) g(u1) coshα0q0(isinhα0),(4. 101) ©200 1 CRC Press LLC Figure 4.12 Electrostatic potential near an oblate spheroidal barrel charged to unit potential with geometrical parameters a/b= 2,β0= 30o.Trun- cation number Ntr= 11. wherem= 1,2,...,u1= cos 2β1, β1=π 2−β0,and x2n=(−1)n 4A2nλ2n(α0)Γ(n+ 1) Γ(n+3 2)h(0,3 2) n−1(u)/braceleftBig h(1,1 2) n−1(u)/bracerightBig−1 2, (4. 102) ε2n= 1−4 λ2n(α0)n(n+1 2)/bracketleftbiggΓ(n+1 2) Γ(n+ 1)/bracketrightbigg2 =O(n−2) asn→ ∞, g(u1) = 1−1 πcoshα0q0(isinhα0)/braceleftBigg/radicalbigg 1 +u1 2+1 2ln/bracketleftBigg√ 2−√1 +u1√ 2 +√1 +u1/bracketrightBigg/bracerightBigg , A0=1 g(u1)/bracketleftBigg 1 +∞/summationdisplay n=1x2nε2nQn(u1)/bracketrightBigg , (4. 103) ©200 1 CRC Press LLC−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−2.5−2−1.5−1−0.500.511.522.5 x/az/a 0.60.70.80.90.920.920.90.80.70.60.5 Figur e4.13 Capacitanc eoftheprolat espheroida lbarrel ,asafunctio nofaspect ratioa/b,forvaryin gapertur esizesβ0. and Qn(u1)=1√π/bracketleftbigg1 2(1+u1)/bracketrightbigg3 2/bracketleftbigg n(n+1 2)/bracketrightbigg−1 2ˆP(0,3 2) n−1(u1). Solvin gthesyste m(4.101)numericall ybythetruncatio nmeth od,and employingtherescalin g(4.102),wemayﬁndthedistributio noftheelectro- staticpotentialneartheconducto rbytheformula(4.13).Anexampl eofthe compute dpotentialnearanoblat espheroida lconducto rwithtwoapertures andcharge dtounitpotentialisshowninFigur e4.12.Theratioofmajorto minor axes is a/b= coshα0/sinhα0= 2 and the angular size of the aperture isβ0= 30o; the system truncation number Ntrwas chosen to be 19. The potential decreases rather uniformly with distance from the structure. ©200 1 CRC Press LLC0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.20.30.40.50.60.70.80.91 R = a / bC β0= 150 β0= 450 β0= 600 β0= 750 Figur e4.14 Capacitanc eoftheoblat espheroida lbarrel ,asafunctio nofaspect ratioa/b,forvaryin gapertur esizesβ0. 4.7Capacitanc eofspheroida lconductors Thesurfac echargedensi tyσaccumulate dontheconducto rsurfac e(α= α0)isdeﬁne dbythejump(4.14)inthenorma lcomponentEαoftheelectric ﬁeldacros sthesurface ,andisgivenbytheexpression s(4.15)and(4.18)for theprolat eandoblat esystems ,respectively.AsnotedinSectio n4.1,thetotal chargeQon each isolated component of the conducting surface is obtained by integration of surface charge density σover the component surface. ©200 1 CRC Press LLC1 1.5 2 2.5 3 3.5 40.511.522.53 R = a / bC β0= 150 β0= 450 β0= 600 β0= 750 Figure 4.15 Capacitance of the slotted prolate spheroidal shell, as a function of aspect ratio a/b, for varying angular size β0of components, each charged to unit potential. 4.7.1 Open spheroidal shells The total charge on an open spheroidal shell comprising a single componentSis, in prolate coordinates, equal to Q=/integraldisplay/integraldisplay SσdS=/integraldisplayπ β=0/integraldisplay2π φ=0σhφhβdφdβ =d 2A(p) 0 Q0(coshα0), (4. 104) or, in oblate coordinates, equal to Q=/integraldisplay/integraldisplay SσdS=/integraldisplayπ β=0/integraldisplay2π φ=0σhφhβdφdβ =d 2A(o) 0 q0(isinhα0). (4. 105) In calculating these integrals we may take the range of βto be [0,π] without aﬀecting the result of integration because, as noted above, the expression for ©200 1 CRC Press LLC0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.10.20.30.40.50.60.70.80.91 R = a / bCβ0= 150 β0= 300 β0= 450 β0= 750 Figur e4.16 Capacitanc eoftheslotte doblat espheroida lshell,asafunctio nof aspectratioa/b,forvaryin gangula rsizeβ0ofcomponents,each charge dtounitpotential. surfac echargedensi tyvanishe sovertheapertur eregion .Werecallthatifthe potentialofanisolate dconducto risequaltounity(U(α0,β)|β∈S0=1),its capacitanc eCandthechargeQarenumericall yequal. Thecapacitanc eC(4.104)oftheprolat espheroida lshellwithtwosym- metrica lcircula rholes(thebarrel )wascompute doverawiderangeofthege- ometrica lparameter sa/bandβ0(thecoeﬃcie ntA(p) 0wasfoundfromFormula (4.81));represe ntativeresult sareprese ntedinFigur e4.13(thegeometrical scaleissetbyb=1,andsod 2=sechα0). Intheoblat ecase,thecapacitanc eC(4.105)wascompute dfrom(4.103); represe ntativeresult sareprese ntedinFigur e4.14(wher ethegeometrica lscale is set byb= 1, and sod 2= cosechα0). We recall that a/bis the ratio of minor to major semi-axes of the prolate spheroid, or the ratio of major to minor semi- axes of the oblate spheroid; in both prolate and oblate systems, β0deﬁnes the angular size of each aperture surface S1(α0,β) :β∈[0,β0]∪[π−β0,π]. The ©200 1 CRC Press LLC1 1.5 2 2.5 3 3.5 40.511.522.53 R = a / bCβ0= 150 β0= 450 β0= 600 β0= 750 capacitanceisanincreasingfunctionofas pectratioandanincreasingfunction ofcom ponentsize. Thetotalchargeonapairofopenspheroidalcapscomposedoftwocompo- nentsS0bothchargedtouni tpotential( U1=U2=1)mayals obecalculated from(4.104)an d(4.105)forth eprolateandoblatecases,res pecti vely. Numericalresultsforth ecapacitanc eCarepresentedi nFigure4.15forthe prolatecas e(th ecoeﬃcie ntA(p) 0isfoundbysolving(4.68))an dinFigure 4.16fortheoblatecas e(thecoeﬃcie ntA(o) 0isfoundfrom(4.93)). β0deﬁnes theangularsizeofth eeachcompone ntofS0(β∈[0,β0]),an db=1.The capacitanceisanincreasingfunctionofas pectrati oandofcapsize. Whenth espheroidalshellwithalongitudinalslotdegeneratestoaclosed spheroidalshell( β0=π 2),thecapacitancesoftheprolateandoblat eclosed shellsobtainedfrom(4.104)and(4.105)ar eexplicitlycalculatedtobe, respecti vely, C(p)=d 21 Q0(coshα0),C(o)=d 21 q0(isinhα0). (4.106) Itiseasytoshowthatthisisidenticalt othatobtaine din[26]byanother method. 4.7.2Spheroidalcondensors Considerthecondensorformedfromoppositelychargedplatesintheform ofspheroidalsegments(Figure4.1(b),4.1(d));th euppe randl owersurfaces are charged to potentials U1= 1 andU2=−1, respectively. The charge Q+ of the positively charged plate is found by the integration of surface chargedensityσ,given in (4. 104) and (4. 105) for the prolate and oblate shells, respectively, over the plate surface S 0=S0(α0,β,ϕ ), where the intervals for integration over β,φare, respectively, [0 ,β0] and [0,2π]. However, we may take the interval for integration over βto be [0,π 2] because over the slot (deﬁned by β∈[β0,π−β0]),the charge equals zero. As a result in the prolate case we obtain, Q+=d 21√π∞/summationdisplay n=0(−1)nA2n+1 4Q2n+1(coshα0)P2n+1(coshα0)Γ(n+1 2) Γ(n+ 2)(4. 107) where {A2n+1}∞ n=0is the solution of the system (4. 68) with l= 1 and employing the rescaling (4. 66); in the oblate case, we obtain Q+=d 21√π∞/summationdisplay n=0(−1)nA2n+1 4q2n+1(isinhα0)p2n+1(isinhα0)Γ(n+1 2) Γ(n+ 2)(4. 108) where {A2n+1}∞n=0is the solution of the system (4. 91) with l= 1 and employing the same rescaling (4. 66). The capacitance of the condensor Cis ©200 1 CRC Press LLC Figur e4.17 Capacitanc eoftheprolat espheroida lcondenser ,asafunctio nof aspectratioa/b,forvaryin gplat esizeβ0. thengivenbytheexpression C=/vextendsingle/vextendsingle/vextendsingle/vextendsingleQ+ U1−U2/vextendsingle/vextendsingle/vextendsingle/vextendsingle. Thecompute dcapacitanc eCofvariou sprolat espheroida lcondensor sis prese ntedinFigur e4.17,whils tthatoftheoblat espheroida lcondensor sis prese ntedinFigur e4.18;β0is the angular size of each capacitor plate. In both cases, the capacitance is an increasing function of aspect ratio and of angular size of the capacitor plates. ©200 1 CRC Press LLC0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.050.10.150.20.250.30.350.40.45 R = a / bCβ0= 150 β0= 450 β0= 600 β0= 750 Figure 4.18 Capacitance of the oblate spheroidal condenser, as a function of aspect ratio a/b, for varying plate sizeβ0. ©200 1 CRC Press LLC1 1.5 2 2.5 3 3.5 400.511.522.533.544.55 R = a / bCβ0= 150 β0= 450 β0= 600 β0= 750 Chapter5 ChargedToroidalShells Toroidalsurfacesprovideani nterestin gcanonicalclas sofconductor sthat illustratemethodsfordeterminin gpote ntialdistribution sandthesurrounding electrostaticﬁeldswhentheyarecharged.Theﬁeldsurroundingaclosed torusanditsassociate dcapacitancehaspreviousl ybee ncalculate d[36,26]. However,ou rinteres tisintheeﬀec tofslotsoraperturesthatmightbeopened inthesurface .Ifsomedegreeofsymmetryi sretained,substanti veprogress withanalyti candsemi-analyti cmeth odscanbemade. Thus ,weﬁrs tconsidercharge dtoroidalconductor swithslotsintroduced sothataxialsymmetryi spreserved,assh owninFigure5.2.Thepote ntial isthe ndeterminedbysolvingdualortripl eseriesequation swit htrigono- metrickernels .Th estandardt oolsprovidedbyth eAbe lintegraltransform approachallowustoregularisetheserie sequationsandcalculatetheelectro- staticpotentia lbysolvin ganinﬁnitesyste moflinearalgebraicequationsof thesecon dkind .Surfacechargedensityan dcapacitanc eoftheseconductors arethe nreadilycomputed .Thematrixoperatorofthissyste misacompletely continuousperturbationofth eidentity(i nthesequencespace l2);thisguar- anteesfastconvergenc eofthetruncate dsystemsolutiontotha tofinﬁnite system,ashasalread ybee ndemonstrate dbysimilarsystemsarisingfrom sphericalan dspheroidalshells(Chapters3and4). Thetoroidalcoordinatesystem( α,β,ϕ )introduce dinSection1.1. 7pro- videsaconvenie ntsystemforformulatin gthepotentialdistributiongenerated byopenchargedtoroidalsurfacesasamixedboundaryvalueproblem.If c isthescalefactor ,thecoordinat esurfaceα=α0deﬁnesatoru swithminor radiusr=ccosechα0andmajorradiu sR=ccothα0, /parenleftBig/radicalbig x2+y2−ccothα/parenrightBig2 +z2=c2cosec2α; itsinteriorandexteriorarerespecti velyspeciﬁedbytheintervals( α0,∞)and [0,α0)forα,whilstβandϕrangeove rtheirfullintervalsofdeﬁnition[ −π,π]. (SeeFigur e5.1. )Inal lournumericalcalculationsthescal efactorcischosen sothatr=1. Weconside rthepotentialdistributionsurroundin gtoroidalsurface swith varioustype sofslotsorapertures .Fixα0andconsiderthetoroidalsurface α=α0(seeFigur e5.2(a)).First,wecalculateth epote ntialsurrounding variousaxiallysymmetricstructure sobtainedbycuttingaxisymmetricslots inthissurface .InSection5.2,th esingleslot(seeFigure5.2(b))isexamined. ©200 1 CRC Press LLC Figur e5.1 Thetoroida lcoordinat esyste mincross-section. Theslotmaybedescri bedbyaﬁxedparamete rβ0;theconnecte dportion ofthetoroida lsurfac egivenbyβ∈[−π,−β0]∪[β0,π]isremovedfromthe complet etorus .Theintroductio noftwotypesof(axisymmetric )slotsis considere dinthefollowingtwosections :transversa lslots(Sectio n5.3,see Figur e5.2(c))thatremovepartoftheconducto rsurfac esothattheremaining segme ntsarespeciﬁe dby α=α0,β∈[−β0,β0]∪[π−β0,π]∪[−π,−(π−β0)], (5.1) andlongitudina lslots(Sectio n5.4,seeFigur e5.2(d)),inwhichthesegme nt(5. 1)isremovedfromthefulltorusα=α0.Capacitance sarebrieﬂ yexamined inSectio n5.5. Thecalculatio nbecome smorecomplicate dwhenaperture sareintroduced sothataxialsymmetr yisbroken.TheﬁnalSectio n(5.6)descri besonesuch structur ethatcanbesolvedsemi-analyticall y–thedegenerat etoroida lshell, withequalmajorandmino rradii,fromwhichanazimuthalsecto risremoved (seeFigur e5.6).Incylindrica lpolars(ρ,θ,φ),thistoroidhasequation (ρ−a)2+z2=a2, ϕ∈[−π,π], ©200 1 CRC Press LLC/0/0 /1/1 /0/0 /1/1/0/0 /1/1 R -c -Rra bβ=const cβ=π β=−πβ=0β0β=β0z x o andtheazimuthalsectorofangularsemi-width φ0, (ρ−a)2+z2=a2,ϕ∈[−φ0,φ0], isremoved.Th epotentialdistributionisdeterminedforthi sstructure ,as wellasforthedegeneratetoroidalsurfacefromwhichmultipleazimuthal sectorsareremove d(seeFigure5.7).Th eapproachinvokestheprincipleof Kelvin inversion (in a sphere) to transform the problem to a set of dual seriesequations dependent upon a continuous spectral parameter. This ﬁnal calculation is a very signiﬁcant extension of analytic and semi- analytic techniques to the determination of the three-dimensional potentialdistribution surrounding nonsymmetric open conducting surfaces. 5.1 Formulation of mixed boundary value problems in toroidal geometry We consider the potential distribution surrounding the toroidal surface α= α0into which one or more axisymmetric slots are introduced; such a surface may be speciﬁed by α=α0, β∈I0, ϕ∈[0,2π], whereI0is a subinterval, or disjoint union of several subintervals of [0 ,2π]. The mixed boundary value problem for the potential theory surrounding sucha slotted toroidal conductor is formulated as follows. Find the function U that is harmonic in R 3, ∆U(α,β,ϕ ) = 0, (5. 2) that satisﬁes the Dirichlet boundary conditions on that part of toroidal surfaceS 0occupied by the conductor, specifying the potential fonS0, U(α0−0,β,ϕ ) =U(α0+ 0,β,ϕ ) =f(β,ϕ),forβ∈I0,ϕ∈[−π,π],(5. 3) that has continuous normal derivative on the aperture surface S1, d dαU(α,β,ϕ )|α=α0+0 α=α0−0= 0,forβ∈[−π,π]\I0,ϕ∈[−π,π], (5. 4) and that vanishes at inﬁnity, U(α,β,ϕ ) =O/parenleftBig |− →r|−1/parenrightBig as|− →r|=/parenleftbig x2+y2+z2/parenrightbig1 2→ ∞, i.e.,Uvanishes as α→0 andβ→0. Also, the electrostatic energy in any volume of space including edges of the conductor must be bounded: W=/integraldisplay/integraldisplay/integraldisplay V|gradU|2dV <∞. (5. 5) ©200 1 CRC Press LLC Intoroidalcoordinates,theLaplaceequation(seeSection1.2.7)admits separationofvariable sandhassolutioninth eform: U(α,β,ϕ )√2coshα−2cosβ= ∞/summationdisplay m=0∞/summationdisplay n=m/bracketleftBig AnmPm n−1 2(coshα)+BnmQm n−1 2(coshα)/bracketrightBig/braceleftbigg cosnβcosmϕ sinnβsinmϕ(5.6) wherePm n−1 2(coshα),Qm n−1 2(coshα)aretoroidalfunctions ,andAnm,Bnmare constantstobedetermine dbythemixe dboundaryconditions .Theseparation constantsn,m areintegersbecause Uisperiodi cinthecoordinate sβand ϕ.Werestrictattentiontoaxisymmetricproblemssothatonlythos eterms withm=0areretaine din(5.6);more over,theopenshellstructurewill beassumedtobesymmetricaboutth exyplane,sothatanyde pendence upontermsi nvolvingsin nβin(5.6)isavoide d(theinterval I0istherefore symmetricaboutth eorigin).Considerin gtheasymptoticbeh aviou rofthe functionsPn−1 2(coshα)andQn−1 2(coshα)atthesingularpoints( α=0,α→ ∞),solutionsofth etype(5.6),whichdecayappropriatel yatinﬁnityan dare continuousacrossth etoroidalsurfac eα=α0,havethefollowingforminthe interior(α≥α0)andexterior(0 ≤α<α 0)regions U(α,β)√2coshα0−2cosβ= ∞/summationdisplay n=0Cncosnβ/braceleftbiggQn−1 2(coshα),α ≥α0, Qn−1 2(coshα0)Pn−1 2(coshα)/Pn−1 2(coshα0),α<α 0. (5.7) Theconstant sCnaretobedetermine dbyenforcementofthemixedboundary conditions(5.3)and(5.4). 5.2Theopenchargedtoroidalsegme nt Thetoroidalshellwithoneslotortoroidalsegme ntisshowninFigure 5.2(b);itoccupiesth eregionα=α0,β∈[−β0,β0] whilst the slot is deﬁned byα=α0,β∈[−π,−β0)∪(β0,π].If the segment is charged to unit poten- tial, enforcement of the boundary conditions (5. 3) and (5. 4) produces the following , ∞/summationdisplay n=0CnQn−1 2(coshα0) cosnβ= (2 coshα0−2 cosβ)−1 2,β∈[0,β0],(5. 8) ©200 1 CRC Press LLC Figur e5.2 Thetorus(a),andvariou storoida lshells :(b)singleslot,(c)two trans versalslotsand(d)twolongitudina lslots. ∞/summationdisplay n=0Cn1 sinhα0Pn−1 2(coshα0)cosnβ=0,β∈(β0,π], (5.9) wherethevalueoftheWronskia nofPn−1 2andQn−1 2(seeAppendix ,(B.69)) has been employed. The toroidal asymmetry factor appearing on the right-hand side of (5. 8) has an expansion in a Fourier series (2 coshα0−2 cosβ)−1 2=1 π∞/summationdisplay n=0(2−δn0)Qn−1 2(coshα0) cosnβ. (5. 10) Substituting (5. 10) in (5. 8) and extracting the zero index terms in (5. 8) ©200 1 CRC Press LLC and(5.9)gi ves ∞/summationdisplay n=1CnQn−1 2(coshα0)cosnβ= /parenleftbigg1 π−C0/parenrightbigg Q−1 2(coshα0)+2 π∞/summationdisplay n=1Qn−1 2(coshα0)cosnβ,β∈[0,β0],(5.11) ∞/summationdisplay n=1Cncosnβ sinhα0Pn−1 2(coshα0)=−C0 sinhα0P−1 2(coshα0),β∈(β0,π].(5.12) Theasymptoticsofth eLegendrefunctionsallowsustoestimate lim n→∞2nsinhα0Pn−1 2(coshα0)Qn−1 2(coshα0)=1;(5.13) wethereforeintroduceth easymptoticallysmal lparameter εn=1−2nsinhα0Pn−1 2(coshα0)Qn−1 2(coshα0)=O(n−2)asn→∞. (5.14) Rescalingth eunkn owns xn=Cn 2nPn−1 2(coshα0)=sinhα0 1−εnCnQn−1 2(coshα0), weconver tEquations(5.8)an d(5.9)totheform ∞/summationdisplay n=1/braceleftbigg xn(1−εn)−2 πsinhα0Qn−1 2(coshα0)/bracerightbigg cosnβ =sin hα0/parenleftbigg1 π−C0/parenrightbigg Q−1 2(coshα0),β∈[0,β0],(5.15) ∞/summationdisplay n=1nxncosnβ=−C0 2P−1 2(coshα0),β∈(β0,π]. (5.16) Thestandardprocedur eforsolvingsu chserie sequationsi nvolvingcosine kernelshasbeendescribe dinSection2. 2insomedetail(se eEquations(2. 39)and(2.40)).Makin gthenecessaryide ntiﬁcationofterms,th esolution maydirectl ybededucedfrom(2.62 )tobeasstate dbel owin(5.21).Let ussketchbrieﬂysom eofth emainstepsinit sdeduction.Itempl oysthe replacementofcosin efunctionsbyJacobipolynomialsgivenby(1.151)and (1.152).Anecessarypreliminar ystepisth eintegrationofbothequationsto increasetheindice softh eJacobipolynomial ssothatth emeth odsofChapter 2areapplicable.Thevariant(2.36)ofRodrigues’formul amaybeapplied ©200 1 CRC Press LLC after the insertion of (1. 151) in (5. 15) and (5. 16); equivalently, we may directly integrate these equations to obtain ∞/summationdisplay n=1/braceleftbigg xn(1−εn)−2 πsinhα0Qn−1 2(coshα0)/bracerightbiggsinnβ n =βsinhα0/parenleftbigg1 π−C0/parenrightbigg Q−1 2(coshα0), β ∈[0,β0],(5. 17) ∞/summationdisplay n=1xnsinnβ= (π−β)C0 2P−1 2(coshα0), β ∈(β0,π]. (5. 18) Settingz= cosβ,z0= cosβ0,and employing the formula (1. 153) produces ∞/summationdisplay n=1xnΓ(n+ 1) Γ(n+1 2)P(1 2,1 2) n−1(z) =C0 P−1 2(coshα0)(1−z2)1 2√π/bracketleftBig arcsinz+π 2/bracketrightBig , z∈(−1,z0),(5. 19) ∞/summationdisplay n=1/bracketleftbigg xn(1−εn)−2 πsinhα0Qn−1 2(coshα0)/bracketrightbiggΓ(n) Γ(n+1 2)P(1 2,1 2) n−1(z) =2 sinhα0 (1−z2)1 2√π/parenleftbigg1 π−C0/parenrightbigg Q−1 2(coshα0)/bracketleftBigπ 2−arcsinz/bracketrightBig , z∈(z0,1). (5. 20) From the Abel integral representation (1. 171) expressing P(1 2,1 2) n−1in terms of P(0,1) n−1,and its companion (1. 172) expressing P(0,1) n−1in terms of P(1 2,1 2) n−1,we derive the inﬁnite system of linear algebraic equations of the second kind for the rescaled unknowns yn=√ 2nxnin the standard way described previously: ym−∞/summationdisplay n=1ynεnQnm(z0) = 2 sinhα0 π∞/summationdisplay n=1√ 2nQn−1 2(coshα0)Qnm(z0)+ 2 sinhα0Q−1 2(coshα0)(1 +z0) πtˆP(0,1) m−1(z0) m.(5. 21) Here t=t1−ln/parenleftbigg1−z0 2/parenrightbigg (5. 22) ©200 1 CRC Press LLC where t1=2sinhα0P−1 2(coshα0)Q−1 2(coshα0), (5.23) Qnm(z0)=/bracketleftBigg ˆQ(1,0) n−1,m−1(z0)+(1+z0)2 tˆP(0,1) n−1(z0)ˆP(0,1) m−1(z0) nm/bracketrightBigg (5.24) and C0=t1 tπ+1 t∞/summationdisplay n=1/bracketleftBigg 1 nynεn+2 π/radicalbigg 2 nsinhα0Qn−1 2(coshα0)/bracketrightBigg × P−1 2(coshα0)(1+z0)ˆP(0,1) n−1(z0).(5.25) NotethatC0isfoundbyenforcementofaconti nuityconditionofthefunction atthepoi ntz=z0. FromthesolutionofthesystemofEquations(5.21),wemayﬁndthe Fouriercoeﬃcientsofth eserie s(5.7)an dthu scalculateth epote ntialU andtheassociate delectrostaticﬁel dnearth echargedtoroidalsegme nt.An exampleisshowninFigur e5.3.Recal lthatth escalefactor cischose nso thattheminorradius r=ccosechα0equals1. 5.3Thetoroidalshellwithtwotrans versa lslots Thissectionbegin stheexaminationoftoroidalsurface swithtwoaxially symmetricslots.Thegeometr yofatoroidalshellwithtwotrans versalslots isshowni nFigure5.2( c).Theconductin gsurfacei sspeciﬁe dby α=α0,β∈[−π,−(π−β0)]∪[−β0,β0]∪[π−β0,π]. If the toroidal segments are charged to unit potential, enforcement on (5. 7) of the boundary conditions (5. 3) (unit potential on the surface), and (5. 4)(continuity of the normal derivative on the slots) leads to the following tripleseries equations with the trigonometric kernels to be solved for the unknowncoeﬃcients C n, ∞/summationdisplay n=0CnQn−1 2(coshα0) cosnβ= (2 coshα0−2 cosβ)−1 2, β∈[0,β0]∪[π−β0,π],(5. 26) ∞/summationdisplay n=0Cn1 sinhα0Pn−1 2(coshα0)cosnβ= 0, β∈(β0,π−β0). (5. 27) ©200 1 CRC Press LLC Figure 5.3 Electrostatic potential surrounding the charged toroidal segment with radii r= 1,R= 2,andβ0= 60o. The property, cos n(π−β) = (−1)ncosnβ,allows us to decouple even and odd index coeﬃcients and obtain the following pair of dual series equationsdeﬁned on the half interval/bracketleftbig 0, π 2/bracketrightbig . The system for the even coeﬃcients is ∞/summationdisplay n=0C2nQ2n−1 2(coshα0) cos 2nβ= 1 2/braceleftBig (2 coshα0−2 cosβ)−1 2+ (2 coshα0+ 2 cosβ)−1 2/bracerightBig , β∈(0,β0) (5. 28) ∞/summationdisplay n=0C2n P2n−1 2(coshα0)cos 2nβ= 0, β ∈(β0,π 2), (5. 29) ©200 1 CRC Press LLC−3 −2 −1 0 1 2 3−3−2−10123 x/az/a 0.60.70.80.850.9 0.95 0.9 0.950.850.80.70.6 whilst that for the odd coeﬃcients is ∞/summationdisplay n=0C2n+1Q2n+1 2(coshα0) cos(2n+ 1)β= 1 2/braceleftBig (2 coshα0−2 cosβ)−1 2−(2 coshα0+ 2 cosβ)−1 2/bracerightBig ,β∈(0,β0) (5. 30) ∞/summationdisplay n=0C2n+1 P2n+1 2(coshα0)cos(2n+ 1)β= 0, β ∈(β0,π 2). (5. 31) Introduce the new variable θ= 2βand setθ0= 2β0.Use the expansions (cf. (5. 10)) (2 coshα0−2 cosβ)−1 2−(2 coshα0+ 2 cosβ)−1 2 =4 π∞/summationdisplay n=0Q2n+1 2(coshα0) cos(n+1 2)θ,(5. 32) (2 coshα0−2 cosβ)−1 2+ (2 coshα0+ 2 cosβ)−1 2 =2 πQ−1 2(coshα0) +4 π∞/summationdisplay n=1Q2n−1 2(coshα0) cosnθ, (5. 33) to obtain the following pair of dual equations deﬁned on the full interval of the variable [0 ,π].The system for the even coeﬃcients is ∞/summationdisplay n=1C2nQ2n−1 2(coshα0) cosnθ =/parenleftbigg1 π−C0/parenrightbigg Q−1 2(coshα0) +2 π∞/summationdisplay n=1Q2n−1 2(coshα0) cosnθ, θ∈(0,θ0), (5. 34) ∞/summationdisplay n=1C2n P2n−1 2(coshα0)cosnθ=−C0 P−1 2(coshα0), θ∈(θ0,π), (5. 35) whilst that for the odd coeﬃcients is ∞/summationdisplay n=0C2n+1Q2n+1 2(coshα0) cos(n+1 2)θ =2 π∞/summationdisplay n=0Q2n+1 2(coshα0) cos(n+1 2)θ, θ∈(0,θ0),(5. 36) ©200 1 CRC Press LLC ∞/summationdisplay n=0C2n+1 P2n+1 2(coshα0)cos(n+1 2)θ= 0, θ ∈(θ0,π). (5. 37) The Equations (5. 34) and (5. 35) are very similar to the equations (5. 8) and (5. 9) considered in the previous section. Setting z= cosθ,z0= cosθ0, we may immediately deduce that the regularised system for the even coeﬃcients is y2m−∞/summationdisplay n=1y2nε2nQnm(z0) = 4 sinhα0 π∞/summationdisplay n=1√ 2nQ2n−1 2(coshα0)Qnm(z0)+ 4 sinhα0Q−1 2(coshα0)(1 +z0) πtˆP(0,1) m−1(z0) m(5. 38) wheret,t1andQnm(z0) are deﬁned by (5. 22)–(5. 24), y2n=C2n√ 2nP2n−1 2(coshα0), C0=t1 tπ+P−1 2(coshα0)(1 +z0) t× ∞/summationdisplay n=1/bracketleftBigg 1 ny2nε2n+4 π/radicalbigg 2 nsinhα0Q2n−1 2(coshα0)/bracketrightBigg ˆP(0,1) n−1(z0), and ε2n= 1−4nsinhα0P2n−1 2(coshα0)Q2n−1 2(coshα0) =O(n−2) asn→ ∞. Let us now turn to the equations (5. 36) and (5. 37). The latter series (5. 37) is nonuniformly convergent and we integrate it to obtain the uniformly convergent series equations ∞/summationdisplay n=0C2n+1 P2n+1 2(coshα0)Γ(n+ 1) Γ(n+3 2)P(1 2,−1 2) n (z) =a√1−z, z∈(−1,z0) (5. 39) ∞/summationdisplay n=0C2n+1Q2n+1 2(coshα0)Γ(n+ 1) Γ(n+1 2)P(−1 2,1 2) n (z) = 2 π∞/summationdisplay n=0Q2n+1 2(coshα0)Γ(n+ 1) Γ(n+1 2)P(−1 2,1 2) n (z),z∈(z0,1) (5. 40) ©200 1 CRC Press LLC where we have replaced sin/parenleftbig n+1 2/parenrightbig θby its representation (1. 154) in terms of the Jacobi polynomial P(−1 2,1 2) n ;ais a constant that will be determined later. Now apply the Abel integral transform technique, employing the integral rep- resentation (1. 172) for P(0,0) n≡Pnin terms of P(1 2,−1 2) n , and the companion representation (1. 171) for Pnin terms of P(−1 2,1 2) n ; from Equations (5. 39) and (5. 40) we may deduce ∞/summationdisplay n=0C2n+1/parenleftbig n+1 2/parenrightbig P2n+1 2(coshα0)Pn(z) =a/radicalbigg 2 πQ−1 2(z), z∈(−1,z0),(5. 41) ∞/summationdisplay n=0C2n+1Q2n+1 2(coshα0)Pn(z) =2 π∞/summationdisplay n=0Q2n+1 2(coshα0)Pn(z), z∈(z0,1). (5. 42) Let x2n+1=C2n+1 2(n+1 2)P2n+1 2(coshα0). As shown previously, the parameter ε2n+1= 1−2(2n+ 1) sinhα0P2n+1 2(coshα0)Q2n+1 2(coshα0) (5. 43) is asymptotically small as n→ ∞ :ε2n+1=O(n−2).The rescaled unknowns satisfy ∞/summationdisplay n=0x2n+1Pn(z) =1 2a/radicalbigg 2 πQ−1 2(z), z ∈(−1,z0), (5. 44) ∞/summationdisplay n=0(1−ε2n+1)x2n+1Pn(z) =sinhα0 π∞/summationdisplay n=0Q2n+1 2(coshα0)Pn(z), z∈(z0,1).(5. 45) Rearranging (5. 41) and (5. 42) gives ∞/summationdisplay n=0x2n+1Pn(z) =/braceleftbigg F1(z), z ∈(−1,z0) F2(z), z ∈(z0,1)/bracerightbigg , (5. 46) where F1(z) =a(2π)−1 2sinhα0Q−1 2(z), F2(z) =∞/summationdisplay n=0x2n+1ε2n+1Pn(z) + 4π−1sinhα0∞/summationdisplay n=0Q2n+1 2(coshα0)Pn(z). ©200 1 CRC Press LLC Theconsta ntaisdetermine dbythecontinuityrequireme ntonthefunction ontheleft-han dsideof(5.46)atthepointz=z0: a=√ 2π sinhα0Q−1 2(z0)∞/summationdisplay n=0/bracketleftbigg x2n+1ε2n+1+4 πsinhα0Q2n+1 2(coshα0)/bracketrightbigg Pn(z0). (5.47) Afterrescalin gtheunkn ownsviay2n+1=/parenleftbig n+1 2/parenrightbig−1 2x2n+1,weobtai nthe followinginﬁnit esystemoflinearalgebrai cequations y2m+1−∞/summationdisplay n=0y2n+1ε2n+1/bracketleftBig ˆQ(0,0) nm(z0)+Rnm(z0)/bracketrightBig = 4 πsinhα0∞/summationdisplay n=0Q2n+1 2(coshα0)(n+1 2)−1 2/bracketleftBig ˆQ(0,0) nm(z0)+Rnm(z0)/bracketrightBig ,(5.48) wherem=0,1,2,..., Rnm(z0)=ˆPn(z0) Q−1 2(z0)Im,Im=/integraldisplayz0 −1Q−1 2(z)ˆPm(z)dz, (5.49) andˆPnisthenormalise dLegendr epolynomial .Theintegral sImarereadily compute d(seeAppendix ,(B.97)): I0=−2/parenleftBig Q1 2(z0)−z0Q−1 2(z0)/parenrightBig , (5.50) Im=z2 0−1 m(m+1)+1 4/parenleftBig Q−1 2(z0)P/prime m(z0)−Q/prime −1 2(z0)Pm(z0)/parenrightBig ,m>0, wherewenotethattheLegendr efunction sQ±1 2aresimpl yexpresse dinterm ofcomplet eellipti cintegral s(seeAppendix ,(B.80)and(B.82)). Thesolutio nofthesystem s(5.38)and(5.48)yieldstheFourie rcoeﬃcie nts oftheseries(5.7),andthusthepotentialandtheassociatedelectrostatic ﬁeldmaybecalculated. Computationall y,thesyste ms(5.38)and(5.48)enjoythesameadvan- tagesastheregularise dsystem sconsider edinChapt er4.Asnotedabove, theLegendr efunction sofhalf-i ntegerindexP±1 2,Q±1 2aresimpl yexpressed intermsofcomplet eellipti cintegral s(seeAppendix ,(B.77)–(B .82)).Recur- rencerelation sforthematri xeleme ntsofthesesystem sarereadil ydeveloped, sonumerica lvaluesoftheunkn ownFourie rcoeﬃcie ntsandtheelectrostatic ﬁeldmaybecompute dveryeﬃcie ntly.Fourcorrec tdigitsinthevaluesof thecoeﬃcie nts{xn}∞ n=0areguara nteedbyachoiceoftruncatio nnumberNtr notexceedin g20.Somecompute dexample softheelectrostati cpotentialare giveninFigure s5.4and5.5forthetoroida lconducto rwithtwotrans versal slots having radii r= 1,R= 2,and angular parameter β0equal to 60◦and 30◦, respectively. ©200 1 CRC Press LLC Figur e5.4 Thecharge dtoroida lshellwithtwotrans versalslots;theradiiare r=1,R=2,andβ0=60o. 5.4Thetoroida lshellwithtwolongitudina lslots Thissectio ncontinuestheexaminatio noftoroida lsurface swithtwoaxially symmetri cslots.Inparticular ,weconside rthesurfac ecompleme ntarytothat ofthepreviou ssection ,wherethelocation sofconductin gsurfac eandslots areinterchange dandconside ratoroida lsurfac ewithlongitudina lslots(see Figur e5.2(d))deﬁne dby α=α0,β∈[−(π−β0),−β0]∪[β0,π−β0], so that the slots occupy the region α=α0,β∈[−π,−(π−β0)]∪[−β0,β0]∪[π−β0,π]. ©200 1 CRC Press LLC−3 −2 −1 0 1 2 3−3−2−10123 x/az/a 0.60.70.80.850.90.950.980.950.90.850.80.70.6 Figure 5.5 The charged toroidal shell with two transversal slots; the radii are r= 1,R= 2,andβ0= 30o. Assume that the segments are charged to unit potential. Then enforcementof the boundary conditions on (5. 7) produces the symmetric triple seriesequations ∞/summationdisplay n=0CnQn−1 2(coshα0) cosnβ= (2 coshα0−2 cosβ)−1 2, β∈[β0,π−β0], (5. 51) ∞/summationdisplay n=0Cn sinhα0Pn−1 2(coshα0)cosnβ= 0, β∈(0,β0)∪(π−β0,π).(5. 52) As in the previous section, these triple series equations may be converted to a decoupled pair of dual series equations for even and odd coeﬃcients. Introducing the new variable θ= 2βand setting θ0= 2β0, the even coeﬃcients ©200 1 CRC Press LLC−3 −2 −1 0 1 2 3−3−2−10123 x/az/a 0.50.60.70.80.850.90.95 0.95 0.95 0.950.90.850.80.70.60.5 satisfy ∞/summationdisplay n=1C2nQ2n−1 2(coshα0) cosnθ= /parenleftbigg1 π−C0/parenrightbigg Q−1 2(coshα0) +2 π∞/summationdisplay n=1Q2n−1 2(coshα0) cosnθ, θ∈(θ0,π), (5. 53) ∞/summationdisplay n=1C2n P2n−1 2(coshα0)cosnθ=−C0 P−1 2(coshα0), θ∈(0,θ0), (5. 54) whilst the odd coeﬃcients satisfy ∞/summationdisplay n=0C2n+1Q2n+1 2(coshα0) cos(n+1 2)θ= 2 π∞/summationdisplay n=0Q2n+1 2(coshα0) cos(n+1 2)θ, θ ∈(θ0,π),(5. 55) ∞/summationdisplay n=0C2n+1 P2n+1 2(coshα0)cos(n+1 2)θ= 0, θ ∈(0,θ0). (5. 56) We ﬁrst consider the system (5. 55)–(5. 56) for the odd coeﬃcients and convert it to dual series equations involving the Jacobi polynomials P(−1 21 2) n withz= cosθ,(z0= cosθ0), ∞/summationdisplay n=0C2n+1 P2n+1 2(coshα0)Γ(n+ 1) Γ(n+1 2)P(−1 2,1 2) n (z) = 0, z∈(z0,1), (5. 57) ∞/summationdisplay n=0C2n+1Q2n+1 2(coshα0)Γ(n+ 1) Γ(n+1 2)P(−1 2,1 2) n (z) = 2 π∞/summationdisplay n=0Q2n+1 2(coshα0)Γ(n+ 1) Γ(n+1 2)P(−1 2,1 2) n (z), z∈(−1,z0).(5. 58) The Abel transform technique may be employed with the integral representa- tion (1. 172) for P(0,0) n≡Pnin terms of P(1 2,−1 2) n,its companion representa- tion (1. 171) for Pnin terms of P(−1 2,1 2) n,and the representation (1. 172) for P(−1 2,1 2) n in terms of Pn.The asymptotically small parameter ε2n+1deﬁned by (5. 43), appears and, arguing as in the last section, we obtain ∞/summationdisplay n=0x2n+1ˆPn(z) =/braceleftbigg0, z ∈(z0,1) F2(z), z∈(−1,z0)/bracerightbigg , (5. 59) ©200 1 CRC Press LLC where F2(z) =∞/summationdisplay n=0x2n+1ε2n+1ˆPn(z) +4 πsinhα0∞/summationdisplay n=0Q2n+1 2(coshα0)ˆPn(z), and the rescaled Fourier coeﬃcients x2n+1=C2n+1/braceleftBigg 2/radicalbigg n+1 2P2n+1 2(coshα0)/bracerightBigg−1 (5. 60) belong tol2.Invoking completeness and orthogonality of the normalised Leg- endre polynomials, we deduce from (5. 59) the following inﬁnite system of linear algebraic equations of the second kind (its matrix operator is a com- pletely continuous perturbation of the identity in l2): (1−ε2m+1)x2m+1+∞/summationdisplay n=0x2n+1ε2n+1ˆQ(0,0) nm(z0) =dm−∞/summationdisplay n=0dnˆQ(0,0) nm(z0), (5. 61) wherem= 0,1,2,..., and dn=4 πsinhα0/radicalbigg n+1 2Q2n+1 2(coshα0). The system (5. 53)–(5. 54) for even coeﬃcients is solved in a similar way, and the rescaled coeﬃcients x2n= (−1)nC2n/braceleftBig√ 2nP2n−1 2(coshα0)/bracerightBig−1 satisfy the i.s.l.a.e. x2m−∞/summationdisplay n=1x2nε2nQnm(z0) = 4 sinhα0 π∞/summationdisplay n=1(−1)n√ 2nQ2n−1 2(coshα0)Qnm(z0)+ 4 sinhα0 πtQ−1 2(coshα0)(1 +z0)ˆP(0,1) m−1(z0) m(5. 62) wherez0= cosθ0,andt,t1andQnm(z0) are deﬁned by (5. 22) and (5. 24), and C0=t1 tπ+P−1 2(coshα0)(1 +z0) t× ∞/summationdisplay n=1/bracketleftBigg 1 nx2nε2n+4 π/radicalbigg 2 nsinhα0Q2n−1 2(coshα0)/bracketrightBigg ˆP(0,1) n−1(z0), ©200 1 CRC Press LLC and ε2n=1−4nsinhα0P2n−1 2(coshα0)Q2n−1 2(coshα0)=O(n−2)asn→∞. Theclose dtoroidalshelli saspeciallimitin gcase.Itcorrespondstosetting β0=π 2in(5.26)and(5.27)fortheconductorwithtransversalslots,orto settingβ0=0in(5.51)and(5.52)fortheconductorwit hlongitudinalslots. Inthesecases,th ecorres pondin gregularise dsystems(5.38),(5.48),or(5. 61),(5.62)havesolutionsinexplici tform. Notingthat ˆQ(1,0) nm(−1)=δnm,thesolutiont o(5.48)with β0=π 2(z0= −1)is y2m+1=/braceleftbigg π(m+1 2)3 2P2m+1 2(coshα0)/bracerightbigg−1 ,m≥0, andthesolutionto(5.38)is y2m=√ 2/braceleftBig π√mP2m−1 2(coshα0)/bracerightBig−1 ,m≥0. Thus C0=π−1,Cn=2π−1,forn=1,2,... (5.63) Substitutingthi ssolutioni n(5.7)produce sapote ntialthatcoincideswith theearlierpublishedsolutionof[36].Ide nticalresultsar eobtainedbysolving thesystems(5.61)and(5.62). Thecomputationalpropertiesofth esystems(5.38)an d(5.48)and(5. 61)–(5.62)arerathe rsimilar ,andasforth etransversalslots,numerical valuesofth eunknownFourierc oeﬃcientsan dtheelectrostati cﬁeldmaybe computedveryeﬃcie ntly,correcttofourdigit swithachoic eoftruncation numberNtrnotexceedin g20. 5.5Capacitanceoftoroidalconductors FollowingthesameargumentasinSection4.7,thecapacitanc eoftheopen toroidal conductor in terms of the Fourier coeﬃcients Cnin (5. 7) is C= 2c/braceleftBigg C0Q−1 2(coshα0) P−1 2(coshα0)+∞/summationdisplay n=1CnQn−1 2(coshα0) Pn−1 2(coshα0)/bracerightBigg . (5. 64) Substitution of the explicit solution (5. 63) for the closed toroidal conductor in (5. 64) produces an expression for capacitance that coincides with thepublished result of [26]. ©200 1 CRC Press LLC Figur e5.6 Adegenerat etoroida lshellwithoneazimuthalcut. 5.6Anopentoroida lshellwithazimuthalcuts Thedeterminatio nofthepotentialdistributio nsurroundin gtheslotte dtor- oidalconductor sconsidere dinpreviou ssection swassigniﬁca ntlyfacilitated bytheiraxialsymmetr y.Thesymmetr ypermitte dtheproble mtobeformu- latedintermsofanappropriat esetofdualortripleseriesequations .The situatio nbecome smorecomplicate dwhenslotsarecutintheshellsothat axialsymmetr yisbroken.Inthissectio nwederivesomenewresult sfora classofconductor swithou taxialsymmetr y,inparticula rfortheperfectly conductin gshellthatispartofadegenerat etorus(inwhichthemajorand mino rradiiareequal )thatmaybeviewedasanincomplet ebodyofrevolution (seeFigur e5.6). Anessentialpreliminar ystepisprovidedbythemeth odofinversionin asphere ,sothatBouwkamp’ stheore m(seeChapte r3)maybeexploited . Someaxiall ysymmetri csituation sarerelati velyeasilyanalyse dbythisap- proach, su chasthespherica lcap(Sectio n3.4).Also,potentialproblem sfor asymmetri cspherica lconductor s(suchastheasymmetri cbarrelorthepair ofasymmetri ccaps)maybesymmetrise dbyaninversionprocesspriorto solutio noftheelectrostati cproble m(Sectio n3.3).More over,theconnec- tion formally described in [77] and [3] between some classes of dual integral equations and dual series equations has the inversion method at its root. Inversion has previously been used for studying charged closed conductors of rather exotic form, such as degenerate tori [7] or spindles [51]. Cutting holes in these surfaces of revolution, without breaking axial symmetry, leads, ©200 1 CRC Press LLC Figur e5.7 Adegenerat etoroida lshellwithfourazimuthalcuts. underinversion ,tothedeterminatio noftheelectrostati cﬁeldproducedby anegati veunitcharge ,locatedontheinversioncentre,inthepresenc eof ﬁniteorsemi-inﬁnit egrounde dcylinder s(inthecaseofthetorus) ,orofopen semi-inﬁnit egrounde dcones(inthecaseofthespindle) .Theseproblem sare there byreduce dtothesolutio nofcertai nwell-studie ddualseriesorintegral equations. Inthissection ,wefocusonconductor swithazimuthalopening sthatbreak theaxialsymmetr y,andthere bydemonstrat eanessentialandsigniﬁca ntex- tensio ntotheclassofthree-dimensiona lopenconductin gsurface swhos epo- tentialisobtainabl ebythesesemi-analyti ctechniques. Thedegenerat etoroida lsurfac eisthebodyofrevolutio ngenerate dbyre- volvin gacircleaboutagiventange nt.Fixin gthistange nttobethez-axisin thecylindrica lcoordinat esyste m(ρ,ϕ,z),andtakin gthecircleradiu stobe aunits ,theclosedsurfac ehastheequatio n(ρ−a)2+z2=a2,ϕ∈[−π,π]. Weﬁrstconside ropentoroida lshellshavingoneazimuthalcut,orhole,of semi-widt hϕ1,speciﬁe dby (ρ−a)2+z2=a2;ϕ∈[−π,−ϕ1]∪[ϕ1,π]. (SeeFigur e5.6.)Subseque ntly,opentoroida lshellswithmultipl eazimuthal cutssymmetricall ydisposedasshowninFigur e5.7willbeexamined. Wewishtodetermin etheelectrostati cpotentialwhensuch openshellsare charge dtounitconsta ntpotential. Le tMdenot etheoriginofthecoordi- natesyste mandconside rinversio nofthetoroida lshellinaspher eofradius 2acentredatM.FromBouwkamp’ stheore m(Sectio n3.4),theproblem is equivalent to the determination of the electrostatic ﬁeld produced by a neg- ©200 1 CRC Press LLC Figur e5.8 (a)Thedegenerat etoroi dwithoneazimuthalcut(topview)and (b)theslotte dinﬁnit ecylinder ,itsimag eunde rinversion. ativeunitcharge ,locatedatM,inthepresenc eofasemi-inﬁnit egrounded cylinde rhavingoneormorelongitudina lslots.(Seeﬁgure5.8.) The equivalent problem may be formulated as a set of dual series equations involving trigonometric functions with unknown Fourier coeﬃcients. However, in contrast to the axially symmetric problems previously investigated, the coeﬃcients depend on some spectral parameter ν. For apertures of arbitrary angle size, regularisation of the dual series equations transforms them to an inﬁnite system of linear algebraic equations of the second kind for the modiﬁedFourier coeﬃcients. The Fredholm nature of the matrix operator, at each ﬁxedvalue of the spectral parameter ν, makes it possible to use a truncation method eﬀectively to obtain a ﬁnite number of Fourier coeﬃcients numerically. An approximate formula for capacitance can be obtained for three limiting cases: the narrow cut ( ϕ 1/lessmuch1 ), a narrow skew ring ( ϕ0=π−ϕ1/lessmuch1), and a large number of cuts ( N/greatermuch1). Some representative numerical results are presented to demonstrate the eﬃcacy of the analysis, and to check theaccuracy of the approximate formulae derived in the limiting cases. 5.6.1 The toroidal shell with one azimuthal cut. Consider ﬁrst the toroidal shell with a single opening arising from an az-imuthal cut. Let Ube the potential associated with the ﬁeld induced by a unit negative charge, located at M, on the inﬁnite circular cylinder of radius 2 a with a longitudinal slot of angular semi-width ϕ 1.The potential must satisfy Laplace’s equation, together with the boundary conditions, edge conditions, ©200 1 CRC Press LLC/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1x yz 2a 2a Mϕ ϕ11x ya) b) andadecayconditionatinﬁnity.Incylindricalc oordinates ,thepote ntial thereforehastheform U(ρ,ϕ,z )=U0+ /integraldisplay∞ 0dνcosνz∞/summationdisplay n=0An(ν)cos(nϕ)/braceleftbigg In(νρ), 0≤ρ<2a In(2νa)Kn(νρ)/Kn(2νa),ρ> 2a (5.65) whereIn,KnarethemodiﬁedBesse lfunctions, U0=−(ρ2+z2)−1 2isthe electrostaticpote ntialofthefreespacenegativeunitchargelocate datM, and{An(ν)}∞ n=0isthesequenceofunknownFouriercoeﬃcients,whi chare functionsofth espectralparameter ν. Usingth emixedboundarycondition sforthesurface U(2a−0,ϕ,z)=U(2a+0,ϕ,z)=0,ϕ∈(ϕ1,π), (5.66) andontheaperture ∂U ∂ρ(2a−0,ϕ,z)=∂U ∂ρ(2a−0,ϕ,z),ϕ∈(0,ϕ1), (5.67) andapplyingtheFourie rcosinetransform,thefollowingdualseriesequations result: ∞/summationdisplay n=0An(ν)cos(nϕ) Kn(2νa)=0,ϕ ∈(0,ϕ1), (5.68) ∞/summationdisplay n=0An(ν)In(2νa)cos(nϕ)=2 πK0(2νa),ϕ∈(ϕ1,π). (5.69) Wenowpr oceed,asusual,totransfor mthisbasi csetofequations,which areofﬁrstkind,toaFredhol mmatrixequationofthesecon dkind.Th emain diﬀerencetopreviou sanalysisi sthatthec oeﬃcie ntsAnarefunction softhe spectralparameter ν. Thestandar dapproachi storeplaceth ecosin ekernelscos( nϕ)byJacobi polynomials P(−1 2,−1 2) n.Itisthennecessarytoi ntegratethes eequations,using thevariant(2.36)Rodrigues’formula,sothatth etechnique sdescribe din Chapter2areapplicable.Equivalently,wemayﬁrs tintegrat ethepair(5. 68) and (5. 69), and then replace the sine kernels by the Jacobi polynomials P(1 2,1 2) n to obtain ∞/summationdisplay n=1An(ν)In(2νa)Γ(n) Γ(n+1 2)P(1 2,1 2) n−1(t) = 2/bracketleftbig 2I0(2νa)A0−4 πK0(2νa)/bracketrightbig √π(1−t2)1 2/parenleftBig arcsint+π 2/parenrightBig , t∈(−1,t1),(5. 70) ©200 1 CRC Press LLC ∞/summationdisplay n=1An(ν) nKn(2νa)Γ(n+ 1) Γ(n+1 2)P(1 2,1 2) n−1(t) = −A0(π−2 arcsint) K0(2νa)√π(1−t2)1 2t∈(t1,1),(5. 71) wheret= cos(ϕ) andt1= cos(ϕ1). Deﬁne the new unknown quantities Mn=An/{nKn(2νa)} that are to be determined. The asymptotics of the modiﬁed Bessel functions (seeAppendix ,(B.159)and(B.160))showthattheparameter εn= 1−2nIn(2νa)Kn(2νa) (5. 72) is asymptotically small: εn=O(n−2) asn→ ∞.The Equations (5. 70) and (5. 71) become /parenleftbig 1−t2/parenrightbig1 2√π∞/summationdisplay n=1MnΓ(n+ 1) Γ(n+1 2)P(1 2,1 2) n−1(t) =−A0 K0(2νa)(π−2 arcsint), t∈(t1,1),(5. 73) /parenleftbig 1−t2/parenrightbig1 2√π∞/summationdisplay n=1Mn(1−εn)Γ(n) Γ(n+1 2)P(1 2,1 2) n−1(t) = 2/bracketleftbigg 2I0(2νa)A0−4 πK0(2νa)/bracketrightbigg/parenleftBig arcsint+π 2/parenrightBig , t∈(−1,t1).(5. 74) The Abel transform method may now be applied. To make the rate of convergence of the terms in series (5. 73) and (5. 74) equal, Equation (5. 73) is integrated using the particular case of (1. 171), (1−t)3 2P(3 2,−1 2) n−1(t) =/parenleftbigg n+1 2/parenrightbigg/integraldisplay1 t(1−x)1 2P(1 2,1 2) n−1(x)dx, (5. 75) to obtain (1−t)3 2√π∞/summationdisplay n=1MnΓ(n+ 1) Γ(n+3 2)P(3 2,−1 2) n−1(t) = A0 K0(2νa)/bracketleftBig 2 (π−2 arcsint) (1 +t)1 2−8 (1−t)1 2/bracketrightBig , t∈(t1,1).(5. 76) Using the Abel-type integral representation (1. 172) for Jacobi polynomials P(1 2,1 2) n−1in terms of P(1,0) n−1and the particular case of (1. 171), P(3 2,−1 2) n−1(t) = (1 −t)−3 2Γ(n+3 2)√πΓ(n+ 1)/integraldisplay1 t(1−x)P(1,0) n−1(x) (x−t)1 2dx, ©200 1 CRC Press LLC a standard argument shows that ∞/summationdisplay n=1Mn(1−t)P(1,0) n−1(t) =/braceleftbigg F1(t), t ∈(−1,t1) F2(t),t∈(t1,1)(5.77) where F1(t) =∞/summationdisplay n=1Mnεn(1−t)P(1,0) n−1(t) + 2/bracketleftbigg 2I0(2νa)A0−4 πK0(2νa)/bracketrightbigg , F2(t) = 2 ln/parenleftbigg1 2(1 +t)/parenrightbigg A0/K0(2νa). A familiar orthogonality argument produces the matrix equation Mm(1−εm) +m∞/summationdisplay n=1MnεnQ(1,0) n−1,m−1(t1) = 2/bracketleftbigg 2I0(2νa)A0−4 πK0(2νa)/bracketrightbigg (1 +t1)P(0,1) m−1(t1)− 2A0 K0(2νa)/bracketleftbigg (1 +t1) ln/parenleftbigg1 +t1 2/parenrightbigg P(0,1) m−1(t1) +1−t1 mP(1,0) m−1(t1)/bracketrightbigg (5. 78) holding for all indices m= 1,2,.... Thesyste m(5.78)hasaninﬁnit enumberofsolution siftheconsta ntA0 has an arbitrary value. A unique solution is obtained by requiring that the function on the left-hand side of (5. 77) is continuous at the point t=t1. Hence A0=4π−1K2 0(2νa)−1 2(1−t1)K0(2νa)/summationtext∞ n=1MnεnP(1,0) n−1(t1) 2I0(2νa)K0(2νa)−ln/parenleftbig1 2(1 +t1)/parenrightbig .(5. 79) A combination of (5. 78) and (5. 79) yields the ﬁnal form of the Fredholm matrix equation of second kind for the unknown Fourier coeﬃcients Mn: Mm(1−εm) +∞/summationdisplay n=1MnεnBnm(t1) =−4 π(1−t1)K0(2νa) 2I0(2νa)K0(2νa)−ln/parenleftbig1 2(1 +t1)/parenrightbigP(1,0) m−1(t1) m(5. 80) wherem= 1,2,..., and Bnm(t1) =n 2/braceleftBigg Q(0,1) n−1,m−1(t1)−(1−t1)2P(1,0) n−1(t1)P(1,0) m−1(t1) nm/parenleftbig 2I0(2νa)K0(2νa)−ln/parenleftbig1 2(1 +t1)/parenrightbig/parenrightbig/bracerightBigg . ©200 1 CRC Press LLC Hereitshouldbenote dthatweuse dtherelationship(cf .(B.172)) Q(1,0) n−1,m−1(t1)=−(1−t1)(1+t1) mP(1,0) n−1(t1)P(0,1) m−1(t1)+n mQ(0,1) n−1,m−1(t1). Thiscompletesth eregularisationoftheoriginalpai rofdualseriesequations (5.68)and(5.69).Computationall y,system(5.80)isveryattractive;it mayberapidlysolvedbyatruncationmeth odwithpredetermine daccuracy forever yvalu eofν,whate vertheangularmeasur eofth eholemaybe.The electrostaticﬁeldisthe nfoun dfrom(5.65)asaFourie rcosinetransfor mof thec oeﬃcie ntsAn(ν). Finally,th ecapacitanceofth econductor ,asafunctionoftheangularsemi- widthϕ1,is C=C(ϕ1)=4a2/integraldisplay∞ 0A0(ν)dν. (5.81) Thelogarithmicsingulari tyofK0aﬀectsthenumericalcalculations ,andthe expressionshouldbetransforme dto C=4a π/integraldisplay∞ 0{I0(x)}−2dx+2a πln/parenleftbigg1+t1 2/parenrightbigg/integraldisplay∞ 0{I0(x)L0(x)}−1dx −a(1−t1)/integraldisplay∞ 0/braceleftBigg∞/summationdisplay n=1Mn(x)εn(x)P(1,0) n−1(t1)/bracerightBigg {L0(x)}−1dx, (5.82) where L0(x)=I0(x)−1 2K0(x)ln/parenleftbigg1+t1 2/parenrightbigg . Thisdependsu pontheidentity(derive dbyanintegrationbyparts) /integraldisplay∞ 0K0(x) I0(x)dx=/integraldisplay∞ 0dx I2 0(x). 5.6.2Thetoroidalshellwithmultipl ecuts Thepote ntialsurroundingatoroidalconductorhaving2N(N=1,2,...)equal azimuthalcutsmaybeanalysedi nthesameway .Thestructurei sdisplayed inFigure5.7.Let ϕ1be the semi-width of each cut: thus ϕ1+ϕ0= 2−Nπ, whereϕ0is the angular semi-width of each of the 2Nconducting sectors. Taking into account the symmetrical location of the cuts and the identity cosnϕ= (−1)ncos(n(π−ϕ)), it is easy to show that the pair of equations corresponding to (5. 68) and (5. 69) take the special form ∞/summationdisplay n=1A2Nn(ν)cos(nθ) K2Nn(2νa)=−A0(ν) K0(2νa), θ∈(0,θ1) (5. 83) ©200 1 CRC Press LLC ∞/summationdisplay n=1A2Nn(ν)I2Nn(2νa) cos(nθ) =2 πK0(2νa)−I0(2νa)A0(ν), θ∈(θ1,π) (5. 84) whereθ= 2Nϕandθ1= 2Nϕ1. Note that Ak= 0 unless kis an integral multiple of 2N. Using the same solution scheme considered above, and introducing the rescaled unknowns M2Nn=A2Nn/{nK2Nn(2νa)}, we obtain the following matrix equation of second kind, M2Nm(1−ε2Nm) +∞/summationdisplay n=1M2Nnε2NnBn,m(u1) =−4(1−u1)K0(2νa)P(1,0) m−1(u1) 2Nmπ/parenleftbig 2I0(2νa)K0(2νa)−ln/parenleftbig1 2(1 +u1)/parenrightbig/parenrightbig,(5. 85) wherem= 1,2,...,u= cosθ,u1= cosθ1,and Bn,m(u1) =n 2Q(0,1) n−1,m−1(u1)− n(1−u1)2P(1,0) n−1(u1)P(1,0) m−1(u1) 2N+1m/parenleftbig 2I0(2νa)K0(2νa)−ln/parenleftbig1 2(1 +u1)/parenrightbig/parenrightbig, A0=4π−1K2 0(2νa)−1 2(1−u1)K0(2νa)/summationtext∞ n=1M2Nnε2NnP(1,0) n−1(u1) 2I0(2νa)K0(2νa)−2−Nln/parenleftbig1 2(1 +u1)/parenrightbig , (5. 86) and ε2Nn= 1−2n.2NI2Nn(2νa)K2Nn(2νa) =O(/parenleftbig 2Nn/parenrightbig−2),asn→ ∞.(5. 87) WhenN= 0 (a single cut), the pairs of Equations (5. 79) and (5. 80) and (5. 85) and (5. 86) are equivalent. It is clear that (5. 85) enjoys the same mathematical and computational properties as obtained for (5. 80), arisingfrom its form as a Fredholm matrix equation of second kind. 5.6.3 Limiting cases The pairs of Equations (5. 79) and (5. 80) and (5. 85) and (5. 86) haveapproximate analytical solutions in three limiting cases: the toroidal surfacewith a narrow single cut ( ϕ 1/lessmuch1 ), the toroidal surface with a large number of cuts (N/greatermuch1), and the narrow skew ring (in which the angle ϕ0=π−ϕ1 satisﬁesϕ0/lessmuch1). ©200 1 CRC Press LLC Whenthecu tinth etorusisnarr ow(ϕ1/lessmuch1,t1=cosϕ1→1),theFourier coeﬃcie ntsofth esystem(5.80)h aveth ebehaviour Mn∼O(ϕ2 1),an dit followsthat C=C0/parenleftbigg 1−1 8ϕ2 1/parenrightbigg +O/parenleftbig ϕ41/parenrightbig , (5.88) where C0=4a π/integraldisplay∞ 0K0(x) I0(x)dx=4a π/integraldisplay∞ 0dx I2 0(x)=1.74138027a (5.89) isthecapacityoftheclosedtoroidalconductor[51].Capacitancevalue sob- tainedfromthisfor mulaagreewellwithresultsofcomputationsonth esystem (5.78),atleas tforcutsofangle ϕ1notexceeding30◦. Whenthetoroidalshel lhasalargenumberofsymmetricallyplacedcuts (N/greatermuch1)itiseas ytosh owthatitscapacitanc eis C=C0/bracketleftbigg 1+1 2Nln/parenleftbigg cosθ1 2/parenrightbigg/bracketrightbigg/bracketleftbig 1+O/parenleftbig 2−2N/parenrightbig/bracketrightbig . (5.90) WhenN→∞ ,expression(5.90)reduce stoth eexpressionforth ecapacity C0ofthefullyclose dconductor.Whe nthecutsarenarrow,for mula(5.90) iscomputationallyveryaccuratebecausebothappr oximationsformultiple holesandfornarr owcutsworktogether. Whentheangularsemi-width ϕ0=π−ϕ1oftheringissmall( ϕ0/lessmuch1), theapproximateexpressionforcapaci tyofthis skew ringis Cring(ϕ0)=4a π/integraldisplay∞ 0K2 0(x) I0(x)K0(2νa)−ln/parenleftbig1 2ϕ0/parenrightbigdx. (5.91) Wemaketworemarksabouttheexpression(5.91).First,ithasalogarithmic singularitynear x=0whichshouldbeaddresse dinanynumericalintegration. Second,theinﬁniterangeofi ntegrationmaybetruncatedto(0 ,4e−γ−1ϕ−1 0) withanerror O(exp/parenleftbig −2ϕ−1 0/parenrightbig ).Valuesofcapacitycomputedaccordingto(5. 91)agreewell,intherange0 <ϕ 0≤10◦,withth enumericalresultsobtained from(5.82)(employin gthesolutionofth esyste m(5.80)). Numericalvaluesforth ecapacityofatoroidalconductorhavingradius a=1andk=2Ncutsmaybeobtainedbysolutionof(5.78)or(5.85) asappropriate.Thesesystemsaretruncatedtoaﬁnitenumbe rofequations and,afte rnumericalsolution,thevalueof A0maybedetermine dfrom(5. 79)o r(5.86)asappropriate .Thecapacity Cisthe ncalculatedaccording to(5.82)byrepeatin gthecalculationfor A0(ν)forasuitablerangeof ν. Selectedresult sareshowninTable5.1( ϕ1istheangularsemi-widthofeach cutindegrees);forsinglecut( k=1),agrap hofcapaci tyCasafunctionof ϕ0=π−ϕ1isshowninFigure5.9.I twasfoundthatthemaximumsizeofa system to be solved did not exceed 10 equations. In the case of a multiply-cut conductor, it was enough to solve only one equation, provided k= 2N≥4. ©200 1 CRC Press LLC Figur e5.9 Capacitanc eofanopentoroida lshellwithazimuthalcuts. Byexaminin gsystem sofrespectiveorder soneandten,thesevendecimal placeresult sdisplayedinTable5.2exemplif y,whenk=4,howtheaccuracy of the computed capacity Ctdepends upon the number tof equations solved after truncation of system (5. 85) to a ﬁnite system. As a consequence, an iteration method may be successfully used to reﬁne accuracy. ©200 1 CRC Press LLC0 10 20 30 40 50 60 70 80 900.40.60.811.21.41.61.82 φ0 (degrees)CAPACITY ϕ1(deg.)k= 1k= 2k= 4k= 16k= 64 0.1 1.741380 1.741379 1.741378 1.741370 1.741338 1.0 1.74131 1.74125 1.74112 1.74032 1.73692 101.7349 1.7285 1.7154 1.5804 301.6893 1.6358 1.5095 901.3912 150 0.9173 170 0.6749 175 0.5800 179 0.4397 179.9 0.3282 179.99 0.26194 Compute dcapaci tyofatoroi dwithkazimutha lcutsofangula rsemi-width ϕ1. ϕ1C1C10 |C10−C1| 3001.5094431 1.5095232 ≈8·10−5 101.7411151 1.7411151<10−7 Compute dcapacitanc evaluesforatoroida lshellwithk=4cuts. ©200 1 CRC Press LLCTable 5.2Table 5.1 Chapter6 PotentialTheor yforConical StructureswithEdges Conicalstructuresar edistinctivelydiﬀerentfromthespheroidalandtoroidal structuresconsideredi npreviouschapters.Electrostati cﬁeldsinduce dbya pointsourceinthevicinityoftheconicaltippossesssingularitie suniqueto thisclass.O ntheotherhand,theope norholl owconicalfrustru mproduced byremovalofth etipregionexhibitsani nterestin grangeofgeometries ,from theﬂat,an nulardis ctoth ehollow,circula rcylinder.Inthi schapter ,wetreat aselectionofpotentialproblemsthataremostdistinctiveofconically-shaped thinconductorswithedges.Theselectioni snotexhaustive ,butisintended toindicatetheclas sofconicalstructuresthatmightbesuccessfullyanalysed bythisapproach. Bywayofintroduction,weﬁrstconside rtherelate dtwo-dimensionalcalcu- lationoftheelectrostati cﬁeldgeneratedbyapairofoppositelychargedstrips thatarenotcoplanarorparallel;th estructureisatwo-dimensionalanalogue oftheconicalfrustrum.Afterconsiderin gtheinﬁnitelylongcone,theelec- trostaticﬁeldofth eopenconicalfrustru misi nvestigatedi nSection6.2.The potentialisdeterminedbyasetofdualintegralequations :anotableas pect oftheirsolutioni stheuseoftheMehler-Focktransforminth eregularisation process .Theresultantsecond-kin dFredholmintegralequationsar ewellcon- ditionedan dposses sthefamiliarpropertie sconducivetothestraightfor ward applicationofstandar dnumericalmeth ods. Thenextsection(6.3)examinesthespindle,whichisth eimageofthe coneunderi nversioni nacentrelocatedo ntheconicalaxis(butnotonthe vertex).Thepote ntialofbothspindl eandcon eareintimatel yrelate dby Bouwkamp’stheorem.Cuttin gasectoralslotinthecon ecorres pond sto openinganazi muthalorlongitudinalslotonth espindlesurface .Bot hstruc- turesareinterestingbecauseofthedeparturefromth eaxialsymmetr yevident inpreviousl yconsidere dconductors.Thedualseriesequationsdescribingthe potentialofth eslotte dconeareregularised;thecapacitanc eoftheass oci- atedslottedspindl eisobtained.Whilstthesepotentialproblem shavebeen studiedpreviously,theirsolutionsar erathe rlesswellknown,es peciallywhen theslotsbreakth eaxialsymmetr yoftheconductor .Asforth etoroidwith azimuthalslotsconsideredinChapter5,thisrepresentsasigniﬁcantextension of analytic and semi-analytic techniques to determining the potential distri- bution surrounding nonsymmetric open conducting surfaces. In this context, ©200 1 CRC Press LLC Figur e6.1 Oppositel ycharge dinﬁnit estrips ,notcoplanar. thehollowspindl ewithaslotisparticularl yinstructi ve,becaus eitusesmost ofthemathematica ltoolssetforthinthisbook. Theﬁnalsectio n(6.4)consider stheconﬂue ntcaseoftheslotte dspindl ein whichtheopenconductin gsurfac ebecome saspherica lshellwithalongitu- dinalslot.Thisclassofnonsymmetri caperture sonthespher ecompleme nts theearlie rstudie sonaxiall ysymmetri copenspherica lconductors. 6.1Non-coplana roppositel ycharge dinﬁnit estrips Letusconside rtheelectrostati cﬁeldduetoapairofoppositel ycharged inﬁnit estripsthatarenotcoplana rorparallel .Thismaybeviewedasa conductin gwedgewithsection sremovedsymmetricall yfromeacharm,as showninFigur e6.1.Thestripslieonhalf-plane semanatin gfromtheorigin and are symmetric with respect to the x-axis. In cylindrical polar coordinates (ρ,ϕ,z ), the positively charged strip is described by ρ∈(a,b),ϕ=ϕ0, and the negatively charged strip by ρ∈(a,b) ;ϕ= 2π−ϕ0.The electrostatic potential ψ(ρ,ϕ,z ) is independent of z; the problem to be solved is two-dimensional, ψ≡ψ(ρ,ϕ). It is convenient to introduce the dimensionless radial coordinate r=ρ/(ab)1 2; Laplace’s equation becomes 1 r∂ ∂r/parenleftbigg r∂ψ(r,ϕ) ∂r/parenrightbigg +1 r2∂2ψ(r,ϕ) ∂ϕ2= 0. (6. 1) ©200 1 CRC Press LLC Thegeometr yofth eproblemforcesustosee kdiscontinuou ssolution sin thevariable ϕ,andimpose sconditionsonth eseparationconstantswhe nthe methodofseparationofvariable sisuse dtoconstructtota lsolution softhe Laplaceequation.Inparticular ,theboundednes softhepote ntialattheorigin andatinﬁnityimplythatineachoftheregions ϕ<ϕ 0andϕ>ϕ 0ithas theform ψ(r,ϕ)=/integraldisplay∞ 0{C(τ)cosτσ+D(τ)sinτσ}/braceleftbig A(τ)e−τϕ+B(τ)eτϕ/bracerightbig dτ, (6.2) where σ=logr, (6.3) andA,B,C, andDareunknownfunctionstobedetermined. Duetothesymmetr yitisclearthat ψ(r,0)=ψ(r,π)=0,r ∈(0,∞). Enforcingaconti nuityconditionat ϕ=ϕ0,thedesiredfor mofsolutionis ψ(σ,ϕ)=/integraldisplay∞ 0dτ{f(τ)cosτσ+g(τ)sinτσ}F(τ,ϕ), (6.4) where F(τ,ϕ)=/braceleftbiggsinh(τϕ),ϕ<ϕ 0 sinh(τϕ0)sinh[τ(π−ϕ)]/sinh[τ(π−ϕ0)],ϕ>ϕ 0 andf,gareunknownfunctionstobedetermined. Themixe dboundaryconditionstobeenforce dontherepresentationare ∂ψ ∂ϕ(σ,ϕ 0−0)=∂ψ ∂ϕ(σ,ϕ 0+0),σ∈(−∞,−σ0)∪(σ0,∞), (6.5) ψ(σ,ϕ 0−0)=ψ(σ,ϕ 0+0)=1,σ∈(−σ0,σ0), (6.6) whereσ0=1 2log(b/a).Itisreadil yjustiﬁe dthatg(τ)≡0,soth eproblem reducestoﬁndingthefunction fthatsatisﬁesthedualintegralequations /integraldisplay∞ 0sinh(τϕ0)f(τ)cos(τσ)dτ=1,σ ∈(0,σ0), (6.7) /integraldisplay∞ 0τsinh(πτ) sinh[(π−ϕ0)τ]f(τ)cos(τσ)dτ=0,σ∈(σ0,∞).(6.8) Whenth estripsarecoplanar/parenleftbig ϕ0=1 2π/parenrightbig ,thepote ntialmaybefoundana- lytically[54,55] .(SeealsoChapter7. )Theparameter ϕ1=1 2π−ϕ0measures the deviation of the structure from the conﬂuent geometry of coplanar strips. ©200 1 CRC Press LLC Toquantifythi seﬀect,letu ssolve(6.7)an d(6.8)byth edeﬁnitionmeth od (Section1.4).Introduceth eauxiliar yfunctiongby /integraldisplay∞ 0τsinh (πτ) sinh [(π−ϕ0)τ]f(τ) cos (τσ)dτ=/braceleftbigg g(σ), σ ∈(0,σ0), 0, σ ∈(σ0,∞).(6. 9) An inverse Fourier cosine transform yields f(τ) =2 πsinh [(π−ϕ0)τ] τsinh (πτ)/integraldisplayσ0 0g(σ/prime) cos (τσ/prime)dσ/prime, (6. 10) and inserting this expression in (6 .7) leads to the ﬁrst-kind Fredholm integral equation for g, /integraldisplayσ0 0g(σ/prime)K(σ,σ/prime)dσ/prime=π 2, σ ∈(0,σ0), (6. 11) where the kernel Kis deﬁned by K(σ,σ/prime) =/integraldisplay∞ 0sinh [(π−ϕ0)τ] sinhτϕ0 τsinh (πτ)cos (τσ) cos (τσ/prime)dτ. (6. 12) One can readily transform Kto a logarithmic type kernel using the cosine- Fourier transform [14] (Vol. 1), valid for |Reα|<π,|Reβ|<π, /integraldisplay∞ 0cosh (βy)−cosh (αy) ysinh (πy)cosxydy =1 2log/bracketleftbiggcoshx+ cosα coshx+ cosβ/bracketrightbigg . (6. 13) By means of some algebraic manipulation K(σ,σ/prime) is transformed to K(σ,σ/prime) = 1 8log/bracketleftBigg (coshσ+ coshσ/prime)2−4/parenleftbig coshσcoshσ/prime+ cos2ϕ1/parenrightbig sin2ϕ1 (coshσ−coshσ/prime)2/bracketrightBigg .(6. 14) The kernel that corresponds to the coplanar structure ( ϕ1= 0) is K0(σ,σ/prime) =1 4log/vextendsingle/vextendsingle/vextendsingle/vextendsinglecoshσ+ coshσ/prime coshσ−coshσ/prime/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (6. 15) Following the basic idea of the method of regularisation, we now split the kernel into two parts, one of which ( K 0) issingular and the other ( K1) is regular , K(σ,σ/prime) =K0(σ,σ/prime) +K1(σ,σ/prime), (6. 16) where K1(σ,σ/prime) =K(σ,σ/prime)−K0(σ,σ/prime) (6. 17) =1 8log/braceleftBigg 1−4coshσcoshσ/prime+ cos2ϕ1 (coshσ+ coshσ/prime)2sin2ϕ1/bracerightBigg . ©200 1 CRC Press LLC Itisevide ntthatthekernelK1isregula rinbothvariable sσandσ/primewithi nthe interval[0,σ0].Theparamete rϕ1measure sthedeviatio nofsolution sfrom thatforcoplana rstrips. Inthecoplana rcase(ϕ1=0)theintegra lequatio nis /integraldisplayσ0 0g0(σ/prime)log/vextendsingle/vextendsingle/vextendsingle/vextendsinglecoshσ+coshσ/prime coshσ−coshσ/prime/vextendsingle/vextendsingle/vextendsingle/vextendsingledσ/prime=2π,σ∈(0,σ0) (6.18) wherefunctio ng0referstothecoplana rstructure .Integra lequation swith logarithmi ckernelsarewellstudied ;manywithcanoni calkernelshaveclosed formsolution s(see,forexample ,[48]).Inparticular ,thesolutio nof(6.18)is g0(σ/prime)=2coshσ0 K(sechσ0)/parenleftbig cosh2σ0−cosh2σ/parenrightbig−1 2,(6.19) whereK(x)denote sthecomplet eellipti cintegra loftheﬁrstkind.Thecor- respondin gfunctio nf0isfoundfrom(6.10),usingtheintegra lreprese ntation (B.100)(seeAppendix ), f0(τ) =coshσ0 K(sechσ0)P−1 2+iτ 2(cosh 2σ0) τcosh/parenleftbigπ 2τ/parenrightbig. (6. 20) The line charge density may now be calculated (recall ρ= (ab)1 2eσ) to be l(ρ) =1 4π/braceleftBig Eϕ/parenleftBig ρ,π 2+ 0/parenrightBig −Eϕ/parenleftBig ρ,π 2−0/parenrightBig/bracerightBig (6. 21) =1 2π/integraldisplay∞ 0τf0(τ) cosh/parenleftBigπ 2τ/parenrightBig cosτσdτ. After substitution of the expression (6 .20) forf0in (6.21) we may use the well-known integral [19] /integraldisplay∞ 0P−1 2+iτ(coshα) cos (τt)dτ=H(α−t)/radicalbig 2 (coshα−cosht), (6. 22) to deduce the line charge density equals (in agreement with [48]) l(ρ) =1 4π.b K(a/b)/braceleftbig/parenleftbig ρ2−a2/parenrightbig/parenleftbig b2−ρ2/parenrightbig/bracerightbig−1 2. (6. 23) In order to examine the potential distribution for non-coplanar strips, we make extensive use of the Mehler-Fock transform [36, 56]. Theorem 6 Letfbe a real valued function deﬁned on the interval (1,∞), which is piecewise continuous and of bounded variation on every ﬁnite subin- terval of (1,∞).Then providing the integrals /integraldisplaya 1|f(x)|(x−1)−3 4dxand/integraldisplay∞ a|f(x)|x−1 2lnx dx ©200 1 CRC Press LLC areﬁniteforeverya>1,therepresentation f(x)=/integraldisplay∞ 0τtanh(πτ)P−1 2+iτ(x)dτ/integraldisplay∞ 1f(ξ)P−1 2+iτ(ξ)dξ (6.24) isvalidateverypointx∈(1,∞)wherefiscontinuous .Thus ,iffhas transform F(τ)=/integraldisplay∞ 1f(ξ)P−1 2+iτ(ξ)dξ, theinvers etransfor mis f(x)=/integraldisplay∞ 0τtanh(πτ)P−1 2+iτ(x)F(τ)dτ. WeshallapplytheMehler- Focktransfor mparticularl yintheform f(α)=1 2/integraldisplay∞ 0τtanh(π 2τ)P−1 2+iτ 2(cosh2α)dτ× /integraldisplay∞ 0f(σ)P−1 2+iτ 2(cosh2σ)sinh(2σ)dσ,(6.25) andusethereprese ntation s(derivedfrom(B.100)oftheAppendix ) P−1 2+iτ 2(cosh 2x) =2 π/integraldisplayx 0cosτt dt/radicalbig cosh2x−cosh2t, =2 πcoth(π 2τ)/integraldisplay∞ xsinτt dt/radicalbig cosh2t−cosh2x.(6. 26) Now integrate (6. 8) to obtain /integraldisplay∞ 0sinh (πτ) sinh [(π−ϕ0)τ]f(τ) sinτσ dτ =C, σ ∈(σ0,∞) , (6. 27) whereCis a constant of integration to be determined. Rescale the function fso that f(τ) =τsinh [(π−ϕ0)τ] sinh (πτ)F(τ). (6. 28) After some manipulation, we obtain the dual integral equations /integraldisplay∞ 0τtanh/parenleftBigπ 2τ/parenrightBig [1−N(τ)]F(τ) cosτσ dτ = 2, σ∈(0,σ0), (6. 29) /integraldisplay∞ 0τF(τ) sinτσ dτ =C, σ∈(σ0,∞) (6. 30) where N(τ) =sinh2τϕ1 sinh2/parenleftbigπ 2τ/parenrightbig,0≤ϕ1<π 2. (6. 31) ©200 1 CRC Press LLC The function Nhas the asymptotic behaviour N(τ) =e−(π−2ϕ1)τ(1 +O(e−2ϕ1τ)), (6. 32) and plays the role of the asymptotically small parameter in the regularisation method; its magnitude is determined by the ratio 2 ϕ1/π.Making use of the integral representations (6. 26), we may obtain the equivalent form /integraldisplay∞ 0τtanh/parenleftBigπ 2τ/parenrightBig [1−N(τ)]F(τ)P−1 2+iτ 2(cosh 2σ)dτ =4 πsechσK(tanhσ), σ ∈(0,σ0),(6. 33) /integraldisplay∞ 0τtanh/parenleftBigπ 2τ/parenrightBig F(τ)P−1 2+iτ 2(cosh 2σ)dτ =2 πCsechσK(sechσ), σ ∈(σ0,∞).(6. 34) We rearrange (6. 33) and (6. 34) so that a suitably chosen singular part may be inverted via the Mehler-Fock transform: Φ(σ) =/integraldisplay∞ 0τtanh/parenleftBigπ 2τ/parenrightBig F(τ)P−1 2+iτ 2(cosh 2σ)dτ =/braceleftbiggF1(σ), σ ∈(0,σ0), F2(σ), σ ∈(σ0,∞),(6. 35) where F1(σ) =/integraldisplay∞ 0τtanh/parenleftBigπ 2τ/parenrightBig N(τ)F(τ)P−1 2+iτ 2(cosh 2σ)dτ +4 πsechσK(tanhσ),(6. 36) F2(σ) =2 πCsechσK(sechσ). (6. 37) The as yet unknown constant Cis determined by invoking the principle that the potential must satisfy the edge condition. This means that the function Φ deﬁned in (6. 35) must be continuous, particularly at σ0,so C=π 2coshσ0 K(sechσ0)/integraldisplay∞ 0τtanh/parenleftBigπ 2τ/parenrightBig N(τ)F(τ)P−1 2+iτ 2(cosh 2σ)dτ + 2K(tanhσ0) K(sechσ0).(6. 38) It will be seen later that this value is exactly the capacitance per unit length of the non-coplanar strips. Before applying the Mehler-Fock transform, it is ©200 1 CRC Press LLC advantageou stoinsertthisvalueforC(6.37),andtoreplac ethecomplete ellipti cintegral sthusoccurrin gin(6.36)and(6.37)bytheirexpressio nin termsofLegendr efunction s(seeAppendix ,(B.88)–(B .92)).Theapplication oftheMehler- Focktransfor mproducesthefollowingsecond-kin dFredholm integra lequatio nforthefunctio nF, F(τ)−/integraldisplay∞ 0F(µ)K(µ,τ)dµ=G(τ),τ∈(0,∞), (6.39) wherethekernelis K(µ,τ)=µ 4tanh/parenleftBigπ 2µ/parenrightBig N(µ)/integraldisplayz0 1P−1 2+iµ 2(z)P−1 2+iτ 2(z)dz+ µ 4tanh/parenleftBigπ 2µ/parenrightBig N(µ)P−1 2+iµ 2(z0) Q−1 2(z0)/integraldisplay∞ z0Q−1 2(z)P−1 2+iτ 2(z)dz,(6.40) withz0=cosh(2σ0),and G(τ)=1 2/integraldisplayz0 1P−1 2(z)P−1 2+iτ 2(z)dz+ 1 2P−1 2(z0) Q−1 2(z0)/integraldisplay∞ z0Q−1 2(z)P−1 2+iτ 2(z)dz.(6.41) Theintegral soccurrin ginthedeﬁnitio nofGmaybeexplicitl yevaluated byusingadiﬀere ntialequatio nfortheLegendr efunction sPν,Qνandtheir Wronskia n(seeAppendix ,(B.63)and(B.69)),sothat G(τ) =2 τ2P−1 2+iτ 2(z0) Q−1 2(z0). (6. 42) In a similar fashion, the explicit closed form for the kernel may be seen to equal K(µ,τ) =µ τ2tanh/parenleftBigπ 2µ/parenrightBig N(µ)/parenleftbig 1−z2 0/parenrightbig P−1 2+iτ 2(z0)P−1 2+iµ 2(z0)× /braceleftbigg q(z0)−µ2pτ(z0)−τ2pµ(z0) µ2−τ2/bracerightbigg ,(6. 43) where q(z0) =Q/prime −1 2(z0) Q−1 2(z0)=/bracketleftbiggd dzlnQ−1 2(z)/bracketrightbigg z=z0, pτ(z0) =P/prime −1 2+iτ 2(z0) P−1 2+iτ 2(z0)=/bracketleftbiggd dzlnP−1 2+iτ 2(z)/bracketrightbigg z=z0. ©200 1 CRC Press LLC Notice that in the limiting case of coplanar strips, the function Nvanishes, and the explicit solution is F(τ) =G(τ) =2 τ2P−1 2+iτ 2(z0) Q−1 2(z0), which coincides exactly with the result stated in Equation (6. 20). When the strips are not coplanar (0 ≤ϕ1<1 2π), the integral Equation (6. 39) may be satisfactorily solved by standard numerical methods. We have implied at various points in this book that the same equations may be solvable in diﬀerent ways. The integral equation (6. 39) should be transformed to some discrete form for this process. At the outset, one mayask if there is a regular basis to construct a satisfactory numerical solution.This question was originally answered aﬃrmatively by C.J. Tranter (see [55]).We describe a similar approach, commencing from (6. 7) and (6. 8). With the rescaling f ∗(τ) =τsinh(πτ) sinh [(π−ϕ0)τ]f(τ), (6. 44) we obtain the dual integral equations /integraldisplay∞ 0M(τ)f∗(τ) cos(τσ)dτ= 1, σ∈(0,σ0), (6. 45) /integraldisplay∞ 0f∗(τ) cos(τσ)dτ= 0, σ∈(σ0,∞), (6. 46) where M(τ) =sinh (τϕ0) sinh [(π−ϕ0)τ] τsinh(πτ)→1,asτ→ ∞. (6. 47) Now represent f∗as a Neumann series f∗(τ) =a0J0(τσ0) + 2∞/summationdisplay n=1√nAnJ2n(τσ0) (6. 48) and substitute in (6. 45) and (6. 46). After interchanging of integration and summation, and using the discontinuous integral /integraldisplay∞ 0J2n(τσ0) cos(τσ)dτ=H(σ0−σ)/parenleftbig σ2 0−σ2/parenrightbig−1 2T2n/parenleftbigg/radicalBig 1−σ2/σ2 0/parenrightbigg , (6. 49) it may be veriﬁed that Equation (6. 46) is satisﬁed automatically, whilst ©200 1 CRC Press LLC Equation (6. 45) leads to 1 2∞/summationdisplay n=1n−1 2AnT2n/parenleftBig/radicalbig 1−z2/parenrightBig (6. 50) = 1−a0/integraldisplay∞ 0M(τ)J0(τσ0) cos(τσ0z)dτ (6. 51) +∞/summationdisplay n=1n1 2An/integraldisplay∞ 0τ−1µ(τ)J2n(τσ0) cos(τσ0z)dτ, (6. 52) wherez∈(0,1) andµ(τ) = 1−2τM(τ) =O/parenleftbig e−2τϕ0/parenrightbig asτ→ ∞.Employing the orthogonality of the Chebyshev polynomials T2nthe Equation (6. 52) is easily transformed to the i.s.l.a.e. of the second kind Am−∞/summationdisplay n=1αnmAn=−αma0, (6. 53) wherem= 1,2,...and a0=/parenleftBigg 1 +∞/summationdisplay n=1βnAn/parenrightBigg /β∗ 0; (6. 54) the coeﬃcients are deﬁned by αnm= 4 (nm)1 2/integraldisplay∞ 0τ−1µ(τ)J2n(τσ0)J2m(τσ0)dτ, αm= 4m1 2/integraldisplay∞ 0M(τ)J0(τσ0)J2m(τσ0)dτ, βn=n1 2/integraldisplay∞ 0τ−1µ(τ)J2n(τσ0)J0(τσ0)dτ, β∗ 0=/integraldisplay∞ 0M(τ)J2 0(τσ0)dτ. (6. 55) Making use of (6. 14), the integral representation for Mis M(τ) =1 2π/integraldisplay∞ 0cos (τz) ln/bracketleftBigg 1 +sin2ϕ0 sinh21 2z/bracketrightBigg dz, (6. 56) so that µ(τ) = 1−τ π/integraldisplay∞ 0cos (τz) ln/bracketleftBigg 1 +sin2ϕ0 sinh21 2z/bracketrightBigg dz. (6. 57) An integration by parts shows that µ(τ) =2 π/integraldisplay∞ 0sin (τz) Φ1(z)dz, (6. 58) ©200 1 CRC Press LLC where Φ1(z) =1 z+1 2d dzln/bracketleftBigg 1 +sin2ϕ0 sinh21 2z/bracketrightBigg =1 z−1 2sin2ϕ0 sinh21 2z+ sin2ϕ0coth/parenleftBigz 2/parenrightBig . (6. 59) Since Φ 1(z)→0 asz→0,we may integrate by parts again to obtain µ(τ) =2 πτ/integraldisplay∞ 0cos (τz) Φ/prime 1(z)dz. (6. 60) This process may be reiterated; it is clear that the asymptotics for µdecrease faster than any power, as µhas exponentially decreasing behaviour. The representations (6. 58) and (6. 60) are satisfactory for numerical calculations of the coeﬃcients αnm,αm,βn,β∗ 0. 6.2 Electrostatic ﬁelds of a charged axisymmetric ﬁnite open conical conductor We have previously exploited solutions of Laplace’s equations in spherical coordinates ( r,θ,ϕ ),which are discontinuous in the radial variable r,to solve various potential problems such as spherical caps. In this chapter we examine ﬁnite open conducting surfaces that are part of the conical surface 0≤r<∞, θ=θ0= constant, 0 ≤ϕ≤2π. In this context, it is necessary to construct the total solution of Laplace’sequation that is discontinuous in the angular variable θ; it is described in [23]. It may then be employed to construct solutions satisfying Dirichlet boundaryconditions on the conductor surface. The form of separated solutions to Laplace’s equations in spherical coor- dinates is given by (1. 62), (1. 63), and (1. 64). Since solutions must beperiodic in ϕ,the separation constant µmust be an integer m≥0, and Φ has the form (a m,bmconstants), Φ (ϕ) =amcosmθ+bmsinmθ. (6. 61) An appropriate choice for the separation constant νappearing in (1. 60) is ν=−1 2+iτ,(τreal) so that the solution for Θ in (1. 60) may be written Θ(θ) =Cm(τ)Pm −1 2+iτ(cosθ) +Dm(τ)Qm −1 2+iτ(cosθ), (6. 62) ©200 1 CRC Press LLC whereCm,Dmaretobedetermined.Thischoiceof νisdictate dbythe requirementthatth eenergyintegral(1.85)isﬁnite;thismaybeveriﬁe dby consideringth eformofth eseparate dsolution sfortheradialc oordinatelisted below(6.65).Alternatively,thesolutionΘmaybeexpressedintermsofthe pairPm −1 2+iτ(cosθ)andPm −1 2+iτ(−cosθ),whichar ealsolinearlyindependent solutionsof(1.60) ;therelation Qm −1 2+iτ(x)=−iπ 2tanh(πτ)Pm −1 2+iτ(x)+(−1)mπ 2sech(πτ)Pm −1 2+iτ(−x) (6.63) isvalidfor |x|≤1.Thus,wesee ksolutionsforΘintheform Θ(θ)=Cm(τ)Pm −1 2+iτ(cosθ)+Dm(τ)Pm −1 2+iτ(−cosθ). (6.64) Theseparate dsolutions(1.62)forth eradialc oordinatetaketh eform R(r)=r−1 2/parenleftbig E(τ)riτ+F(τ)r−iτ/parenrightbig =e−σ 2(e(τ)cos(τσ)+f(τ)sin(τσ)), (6.65) whereσ=lnr.Theseparatedsolution ψτ m(r,θ,ϕ )forLaplace’sequation correspondin gtoparameters mandτistheproductof(6.62),(6.64),and (6.65),an dthegeneralsolutionisthesuperposition ψ(r,θ,ϕ )=∞/summationdisplay m=−∞/integraldisplay∞ −∞ψτ m(r,θ,ϕ )dτ. (6.66) SolutionsofLaplace’ sequation sinregionsboundedbyth econicalsurface θ=θ0musttakeaccountofthesingularbehaviou rofth eassociatedLegendre functionsin(6.64)atthesingularpoints θ=0,π.Thefunction Pm −1 2+iτ(cosθ) isboundedat θ=0,yetu nbounde datθ=π.Thus internal conicalharmon- icsintheregion0 ≤θ≤θ0involvePm −1 2+iτ(cosθ)(an dDm(τ)=0in(6. 64)),whereas external conicalharmonicsintheregion θ0≤θ≤πinvolve Pm −1 2+iτ(−cosθ)(an dCm(τ)=0).Iftheconductorisformedfromthebi- conicalsurface θ=θ0,θ=θ1,conicalharmonicsforth eintermediateregion θ0≤θ≤θ1employbot hterm sin(6.64). Letusconsiderth eDiri chletboundaryvalueproble mforLaplace’sequation foranope nhollowﬁniteconicalconductorofthety pesh owninFigure6.2. The boundary condition on the single cone θ=θ0,or on the frustrum lying on this surface, is ψ(σ,θ0−0,ϕ) =ψ(σ,θ0+ 0,ϕ),− ∞<σ< ∞,0≤ϕ≤2π. (6. 67) ©200 1 CRC Press LLC Figure 6.2 Various conical structures: (a) the bicone, (b) the cone, (c) a pair of hollow conical frustra, and (d) a hollow conical frustrum of ﬁnite length. The particular solution (6. 64) takes the form Θ (θ) = Cm(τ)/braceleftBigg Pm −1 2+iτ(cosθ), θ ∈(0,θ0), Pm −1 2+iτ(−cosθ)Pm −1 2+iτ(cosθ0)/Pm −1 2+iτ(cosθ0), θ∈(θ0,π). (6. 68) For the ﬁnite hollow biconical conductor lying on the bicone θ=θ0,θ=θ1, the particular solution (6. 64) takes the form Θ (θ) =  A m(τ)Pm −1 2+iτ(cosθ), θ ∈(0,θ0), ∆−1 m(τ)/bracketleftBig ∆(1) m(τ)Pm −1 2+iτ(cosθ) + ∆(2) m(τ)Pm −1 2+iτ(−cosθ)/bracketrightBig , θ∈(θ0,θ1), Dm(τ)Pm −1 2+iτ(−cosθ), θ ∈(θ1,π), (6. 69) ©200 1 CRC Press LLC where ∆m(τ) =Pm −1 2+iτ(cosθ0)Pm −1 2+iτ(−cosθ1) −Pm −1 2+iτ(−cosθ0)Pm −1 2+iτ(cosθ1), ∆(1) m(τ) =Am(τ)Pm −1 2+iτ(−cosθ1)Pm −1 2+iτ(cosθ0) −Dm(τ)Pm −1 2+iτ(−cosθ1)Pm −1 2+iτ(−cosθ0), ∆(2) m(τ) =−Am(τ)Pm −1 2+iτ(cosθ0)Pm −1 2+iτ(cosθ1) +Dm(τ)Pm −1 2+iτ(cosθ0)Pm −1 2+iτ(−cosθ1). (6. 70) The fundamental solution of Laplace’s equation is the inverse distance func- tion/vextendsingle/vextendsingle/vextendsingle− →r−− →r/prime/vextendsingle/vextendsingle/vextendsingle−1 given by (1. 209); in terms of the notation introduced it takes the form /vextendsingle/vextendsingle/vextendsingle− →r−− → r/prime/vextendsingle/vextendsingle/vextendsingle−1 =e−1 2(σ+σ/prime){2 (cosh (σ−σ/prime)−cosψ)}−1 2, (6. 71) whereσ= lnr, σ/prime= lnr/primeand cosψ= cosθcosθ/prime+ sinθsinθ/primecos(φ−φ/prime). It is representable as the integral transform [23] /vextendsingle/vextendsingle/vextendsingle− →r−− → r/prime/vextendsingle/vextendsingle/vextendsingle−1 =e−1 2(σ+σ/prime)/integraldisplay∞ 0sech (πτ)P−1 2+iτ(cosψ) cosτ(σ−σ/prime)dτ. (6. 72) Employing the addition formula for the Legendre function [1] Pν(cosxcosy±sinxsinycosa) =∞/summationdisplay k=0(±1)kcoskaPk ν(cosx)P−k ν(cosy), (6. 73) wherex≥0, y<π, andx+y<π, and the relation [1] P−k ν(y) = (−1)kΓ (ν−k+ 1) Γ (ν+k+ 1)Pk ν(y),|y|<1, (6. 74) we ﬁnally obtain the representation which is discontinuous in θ: /vextendsingle/vextendsingle/vextendsingle− →r−− → r/prime/vextendsingle/vextendsingle/vextendsingle−1 =e−1 2(σ+σ/prime)∞/summationdisplay m=0(2−δ0m) cosm(φ−φ/prime)× /integraldisplay∞ 0sech (πτ)Γ/parenleftbig1 2+iτ−m/parenrightbig Γ/parenleftbig1 2+iτ+m/parenrightbigP(θ,θ/prime) cosτ(σ−σ/prime)dτ(6. 75) where P(θ,θ/prime) =/braceleftBigg Pm −1 2+iτ(−cosθ)Pm −1 2+iτ(cosθ/prime), θ<θ/prime Pm −1 2+iτ(cosθ)Pm −1 2+iτ(−cosθ/prime), θ>θ/prime/bracerightBigg . ©200 1 CRC Press LLC The function (6. 75) describes the potential generated by an elementary point charge located at the point ( r/prime,θ/prime,ϕ/prime).When it is located on the z-axis at points with θ/prime= 0,π,this expression simpliﬁes because Pm −1 2+iτ(1) =δ0m, to ψ0(σ,θ) =e−1 2(σ+σ/prime)/integraldisplay∞ 0sech (πτ)P−1 2+iτ(∓cosθ) cosτ(σ−σ/prime)dτ, (6. 76) where the minus (respectively plus) sign refers to the choice θ/prime= 0 (respec- tivelyπ). Let us consider the simplest problem, the earthed semi-inﬁnite cone θ=θ0 in the presence of an elementary positive charge located on the z-axis atσ= σ/prime,θ/prime= 0.This is a standard internal boundary value problem with Dirichlet boundary conditions given on the conical surface. The total electrostatic potentialψis sought as the sum ψ=ψ0+ψ1 of the primary potential ψ0given by (6. 76) and an induced potential ψ1, subject to the boundary condition ψ(σ,θ0) = 0,− ∞<σ< ∞. (6. 77) The induced potential ψ1is constructed as a superposition of internal conical harmonics ψ1(σ,θ) =e−1 2(σ+σ/prime)/integraldisplay∞ 0sech (πτ)f(τ)P−1 2+iτ(cosθ) cosτ(σ−σ/prime)dτ, (6. 78) where the function fis found by the boundary condition to be f(τ) =−P−1 2+iτ(−cosθ0) P−1 2+iτ(cosθ0). (6. 79) The surface charge density S(σ) is easily deduced to be S(σ) =/bracketleftbigg −e−σ∂ ∂θψ(σ,θ)/bracketrightbigg θ=θ0 =2 πe−1 2(3σ+σ/prime)cosecθ0/integraldisplay∞ 0cosτ(σ−σ/prime) P−1 2+iτ(cosθ0)dτ. (6. 80) Whenθ0=1 2π,the cone degenerates to the plane z= 0.Using the value [36] P−1 2+iτ(0) =√π/vextendsingle/vextendsingle/vextendsingle/vextendsingleΓ/parenleftbigg3 4+iτ 2/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle−2 , (6. 81) ©200 1 CRC Press LLC andthetabulate dintegra l[14] /integraldisplay∞ 0|Γ(a+ix)|2cosxydx=πΓ(2a)2−2asech2a(y/2),(a,y>0),(6.82) wemayverifythatthesurfac echargedensi tyS(ρ)duetothepointchargelo- catedonthez-axisatadistanc edfromtheoriginofthecylindrica lcoordinate syste mis S(ρ)=2d/parenleftbig ρ2+d2/parenrightbig−3 2; thisisinaccor dwithresult sobtaine dbyeleme ntarymeth ods. Whenθ0/lessmuch1,thestructur ebecome saverysharphollowcone.Inthiscase itisconvenienttousethehypergeometri creprese ntation(seeAppendix ,(B. 98)) P−1 2+iτ(cosθ0) = 2F1/parenleftbigg1 2+iτ,1 2−iτ; 1; sin21 2θ0/parenrightbigg = 1 +/parenleftbigg1 4+τ2/parenrightbigg sin21 2θ0+O/parenleftbigg sin41 2θ0/parenrightbigg .(6. 83) Inserting this approximation in (6. 80) produces the approximation for surface charge density S(σ)/similarequal1 r3 2r/prime1 2cosecθ0cosec1 2θ0/parenleftBigr r/prime/parenrightBig2/θ0 , r<r/prime. (6. 84) ThusS(σ)→0 asr→0. If the elementary charge is located outside the cone (at σ=σ/prime,θ/prime=π), then Equation (6. 80) is replaced by S(σ) =2 πe−1 2(3σ+σ/prime)cosecθ0/integraldisplay∞ 0cosτ(σ−σ/prime) P−1 2+iτ(−cosθ0)dτ. (6. 85) For the same limiting structure of the very sharp hollow cone ( θ0/lessmuch1),the approximation P−1 2+iτ(−cosθ0)/similarequal2 πcosh(τπ) ln/parenleftbigg2 θ0/parenrightbigg (6. 86) is relevant and S(σ)/similarequal(r+r/prime)−1 rcosecθ0 ln (2/θ0). (6. 87) The approximation (6. 87) has a singularity at the vertex of the cone. This phenomenon is discussed in some detail by Hobson [23]. Remove the lower and upper parts of the inﬁnite hollow cone to form the ﬁnite hollow frustrum a≤r≤b, θ=θ0, ©200 1 CRC Press LLC showni nFigure6.2.Itisconvenie nttointroducethenormalise dradialcoor- dinate ρ=r/√ ab, so that the frustrum is given by ρ−1 0≤r≤ρ0, θ=θ0, whereρ0= (b/a)1 2.Also replace the variable σ= lnrpreviously introduced by σ= lnρ,so the frustrum is described by σ∈(−σ0,σ0),whereσ0=1 2ln (b/a). The partial solutions for the variable σmay still be described by Equation (6. 65). Accordingly, let us consider the Dirichlet boundary value problem for the frustrum. The axisymmetric potential ψ(σ,θ) must satisfy the mixed boundary conditions ∂ ∂θψ(σ,θ0−0) =∂ ∂θψ(σ,θ0+ 0), σ∈(−∞,−σ0)∪(σ0,∞) (6. 88) ψ(σ,θ0−0) =ψ(σ,θ0+ 0) = Φ(σ), σ∈(−σ0,σ0), (6. 89) where Φ is the given value of the potential on the conductor. We consider the simplest case where the frustrum is charged to unit potential (Φ = 1). Usingthe superposition (6. 66), the solution for ψis sought in the form ψ(σ,θ) =e −σ 2/integraldisplay∞ 0dτ{fc(τ) cosτσ+fs(τ) sinτσ}F(τ,θ), (6. 90) where F(τ,θ) =/braceleftbiggP−1 2+iτ(cosθ), 0≤θ≤θ0, P−1 2+iτ(cosθ0)P−1 2+iτ(cosθ)/P−1 2+iτ(−cosθ0), θ0≤θ≤π andfc,fsare unknown functions to be determined. Because of the symmet- ric subdivision of the real line, the integral equations for fcandfscan be decoupled to the following sets of dual integral equations: /integraldisplay∞ 0fc(τ)P−1 2+iτ(cosθ0) cosτσdτ = cosh(σ/2), σ∈(0,σ0), (6. 91) /integraldisplay∞ 0fc(τ)cosh(πτ) P−1 2+iτ(−cosθ0)cosτσdτ = 0, σ∈(σ0,∞), (6. 92) and /integraldisplay∞ 0fs(τ)P−1 2+iτ(cosθ0) sinτσdτ = sinh(σ/2), σ∈(0,σ0), (6. 93) /integraldisplay∞ 0fs(τ)cosh(πτ) P−1 2+iτ(−cosθ0)sinτσdτ = 0, σ∈(σ0,∞). (6. 94) ©200 1 CRC Press LLC Whenθ0=1 2π,theseequation sdescri bethenontrivia lgeometr yofthe charge dannulardisc[55]. Theﬁrststepistointegrat e(6.92)andtodiﬀere ntiate(6.93),respectively, obtaining /integraldisplay∞ 0fc(τ)cosh(πτ) τP−1 2+iτ(−cosθ0)sinτσdτ=C,σ∈(σ0,∞), (6.95) whereCisaconsta ntofintegratio ntobedetermine dand /integraldisplay∞ 0τfs(τ)P−1 2+iτ(cosθ0)cosτσdτ=1 2cosh(σ/2),σ∈(0,σ0).(6.96) Rescal etheunkn ownfunctions fc(τ)=τ2sech(πτ)P−1 2+iτ(−cosθ0)Fc(τ), (6.97) fs(τ)=τsech(πτ)P−1 2+iτ(−cosθ0)Fs(τ), (6.98) sothat /integraldisplay∞ 0τtanh(πτ)M(τ;θ0)Fc(τ)cosτσdτ=cosh(σ/2),σ∈(0,σ0)(6.99) /integraldisplay∞ 0τFc(τ)sinτσdτ=C,σ∈(σ0,∞), (6.100) and /integraldisplay∞ 0τtanh(πτ)M(τ;θ0)Fs(τ)cosτσdτ=1 2cosh(σ/2),σ∈(0,σ0)(6.101) /integraldisplay∞ 0τFs(τ)sinτσdτ=0,σ∈(σ0,∞), (6.102) wherethefunctio nMisdeﬁne dby M(τ;θ0)=τcosech(πτ)P−1 2+iτ(cosθ0)P−1 2+iτ(−cosθ0). (6.103) Theasymptotic sfortheconica lfunction s(seeAppendix ,(B.101))showthat lim τ→∞M(τ;θ0) =1 πcosecθ0, and the function N(τ;θ0) = 1−πsinθ0M(τ;θ0) = 1−πτsinθ0 sinh(πτ)P−1 2+iτ(cosθ0)P−1 2+iτ(−cosθ0) =O(τ−2) asτ→ ∞. (6. 104) ©200 1 CRC Press LLC Equations (6. 99) and (6. 100) may be rearranged in the form Φc(σ) =/integraldisplay∞ 0τtanh(πτ)Fc(τ)P−1 2+iτ(coshσ)dτ =/braceleftbigg F1(σ), σ ∈(0,σ0), F2(σ), σ ∈(σ0,∞),(6. 105) where F1(σ) =πsinθ0+/integraldisplay∞ 0τtanh(πτ)Fc(τ)N(τ;θ0)P−1 2+iτ(coshσ)dτ, F2(σ) =2 πCQ−1 2(coshσ). Likewise, (6. 101) and (6. 102) may be rearranged in the form Φs(σ) =/integraldisplay∞ 0τtanh(πτ)Fs(τ)P−1 2+iτ(coshσ)dτ =/braceleftbiggF3(σ), σ ∈(0,σ0), 0, σ ∈(σ0,∞),(6. 106) where F3(σ) =π 2sinθ0+/integraldisplay∞ 0τtanh(πτ)Fs(τ)N(τ;θ0)P−1 2+iτ(coshσ)dτ. Note that in deriving (6. 105), we also used the relation (B. 89) of the Ap- pendix. As shown previously, the value of the constant Cis determined by enforcing the requirement of continuity on Φ catσ0,giving C=π 2/integraldisplay∞ 0τtanh(πτ)Fc(τ)N(τ;θ0)P−1 2+iτ(coshσ0) Q−1 2(coshσ0)dτ +π2 2sinθ0 Q−1 2(coshσ0).(6. 107) The ﬁnal step uses the inverse of the Mehler-Fock transform to convert both (6. 105) and (6. 106) to second-kind Fredholm integral equations to be solved for the functions FcandFs, respectively, obtaining Fs(τ) =/integraldisplay∞ 0Fs(ξ)Ks(ξ,τ)dξ=Gs(τ) (6. 108) and Fc(τ) =/integraldisplay∞ 0Fc(ξ)Kc(ξ,τ)dξ=Gc(τ). (6. 109) The inhomogeneous terms are Gs(τ) =π 2sinθ0 1 4+τ2/parenleftbig 1−z2 0/parenrightbig P/prime −1 2+iτ(z0), (6. 110) ©200 1 CRC Press LLC Gc(τ)=τ−2πsinθ0/parenleftbig 1−z2 0/parenrightbig P−1 2+iτ(z0)× /braceleftBiggQ/prime −1 2(z0) Q−1 2(z0)−1 1+4τ2P/prime −1 2+iτ(z0) P−1 2+iτ(z0)/bracerightBigg ,(6.111) andthekernels ,respectively,aredeﬁne dtobe Ks(ξ,τ)=ξtanh(πξ)N(ξ;θ0)R(ξ,τ;z0), (6.112) Kc(ξ,τ)=ξtanh(πξ)N(ξ;θ0)R(ξ,τ;z0) +ξtanh(πξ)N(ξ;θ0)P−1 2+iξ(z0) Q−1 2(z0)R∗(τ;z0),(6.113) where R(ξ,τ;z0)=/integraldisplayz0 −1P−1 2+iξ(z)P−1 2+iτ(z)dz, R∗(τ;z0)=/integraldisplay∞ z0Q−1 2(z)P−1 2+iτ(z)dz. BothRandR∗maybereadil yevaluate dinclosedform(inthesamewayas Formula(B.97)oftheAppendix ): R(ξ,τ;z0) =/parenleftbig 1−z2 0/parenrightbig (τ2−ξ2)/braceleftBig P−1 2+iξ(z0)P/prime −1 2+iτ(z0)−P/prime −1 2+iξ(z0)P−1 2+iτ(z0)/bracerightBig ,(6. 114) R∗(τ;z0) = /parenleftbig 1−z2 0/parenrightbig τ2/braceleftBig P−1 2+iτ(z0)Q/prime −1 2(z0)−P/prime −1 2+iτ(z0)Q−1 2(z0)/bracerightBig .(6. 115) For an eﬀective numerical solution, the kernels of (6. 108) and (6. 109) must converge suﬃciently fast as ξ→ ∞.This depends completely on the asymptotics of the function N.It is convenient to examine the function N∗(τ;θ0) = tanh(πτ)N(τ;θ0) = (tanh(πτ)−1) +N∗ 1(τ;θ0), (6. 116) where N∗ 1(τ;θ0) = 1−πτsinθ0 cosh (πτ)P−1 2+iτ(cosθ0)P−1 2+iτ(−cosθ0).(6. 117) Since tanh( πτ)−1 =−2e−2πτ/parenleftbig 1 +O/parenleftbig e−2πτ/parenrightbig/parenrightbig asτ→ ∞,it is quite clear that the major contribution to the asymptotic behaviour of N∗is dominated by ©200 1 CRC Press LLC that ofN∗ 1.The asymptotic behaviour of N∗ 1can be determined by examining an integral representation for the product of conical functions appearing in (6. 117). Integrate both sides of (6. 75) twice. Assuming σ/prime= 0,ϕ/prime= 0,θ/prime=θ0, ﬁrst integrate w.r.t. ϕover the interval [0 ,2π].Multiply the result of the ﬁrst integration by cos µσand integrate w.r.t. σover the interval (0 ,∞) to obtain 1√ 2/integraldisplay∞ 0dσcosτσ/integraldisplay2π 0{coshσ−cosθcosθ0−sinθsinθ0cosϕ}−1 2dϕ =π2sech(πτ)/braceleftbiggP−1 2+iτ(−cosθ)P−1 2+iτ(cosθ0), θ<θ 0 P−1 2+iτ(cosθ)P−1 2+iτ(−cosθ0), θ>θ 0.(6. 118) Using the tabulated integral [14] /integraldisplayπ 0dx√ a±bcosx=2√ a+bK/parenleftBigg/radicalbigg 2b a+b/parenrightBigg , a>b> 0, (6. 119) and the relation Q−1 2(1 + 2x2) =1√ 1 +x2K/parenleftbigg1√ 1 +x2/parenrightbigg , (6. 120) we obtain the desired result 2 π2cosh(πτ)√sinθsinθ0/integraldisplay∞ 0Q−1 2/parenleftbiggcoshz−cosθcosθ0 sinθsinθ0/parenrightbigg cosτz dz =/braceleftbiggP−1 2+iτ(−cosθ)P−1 2+iτ(cosθ0), θ<θ 0 P−1 2+iτ(cosθ)P−1 2+iτ(−cosθ0), θ>θ 0.(6. 121) Settingθ=θ0, the value of N∗ 1(τ;θ0) is deduced to be N∗ 1(τ;θ0) = 1−2τ π/integraldisplay∞ 0Q−1 2/parenleftbigg 1 + 2sinh2(z/2) sin2θ0/parenrightbigg cosτz dz. (6. 122) After a single integration by parts this may be written as N∗ 1(τ;θ0) =2 π/integraldisplay∞ 0S1(z) sinτz dz, (6. 123) where S1(z) =1 z+d dzQ−1 2/parenleftbigg 1 + 2sinh2(z/2) sin2θ0/parenrightbigg . SinceS1(z)/revsimilarzlnzwhenz/lessmuch1,andS1(z)→0 asz→0,another integration by parts produces N∗ 1(τ;θ0) =2 πτ/integraldisplay∞ 0S2(z) cosτz dz (6. 124) whereS2(z) =S/prime 1(z); it may now be deduced that N∗ 1(τ;θ0) =O(τ−2) as τ→ ∞. Thus, standard methods for second-kind Fredholm equations may be em- ployed eﬀectively to obtain numerical solutions to (6. 108) and (6. 109). ©200 1 CRC Press LLC Figur e6.3 Thespindle .Thesemi-inﬁnit econeisitsimag eunde raninversion withcentreA. 6.3Theslotte dhollowspindle Theelectrostati cpotentialofseveralstructure srelate dtothespindlecan bededuce dfromthesolution salread yobtaine dforconica lstructures .The basicstructur eofthespindl eisthesurfac eofrevolutio nobtaine dbyrevolving anarcofacircleaboutthechordOAjoinin gitsendpointsAandO.The vector−→OAmaybechosentoliealongthepositivedirectio nofthez-axis;the coordinat eoriginmaybelocatedatO.(SeeFigur e6.3.) Unde rinversioninthesphere of radius R=OAcentredatA,theimage ofthespindl eistheinﬁnit e(rightcircular) con ewithvertexOandaxis coincidin gwiththez-axis.Thehalf-angl eαoftheconeequals half the angle subtended by the chord OAatthecentreof itscircle . Bouwkamp’ stheore m(seeChapte r3)maybeused to calculate the capac- ©200 1 CRC Press LLC itance of the spindle (see, for example, [51]). The calculation is equivalent to the calculation of the electrostatic ﬁeld surrounding the grounded semi-inﬁnite cone in the presence of a unit negative charge located at A; in the usual spherical coordinates the charge is located at ( r,θ,φ ) = (R,π, 0). The potential ψ 0due to this charge is given by (6. 76), with σ= lnrand σ/prime= lnR, ψ0(σ,θ) =−e−1 2(σ+σ/prime)/integraldisplay∞ 0sech (πτ)P−1 2+iτ(cosθ) cosτ(σ−σ/prime)dτ. (6. 125) The induced potential has the form ψ1(σ,θ) =e−1 2(σ+σ/prime)/integraldisplay∞ 0sech (πτ)f(τ)P−1 2+iτ(−cosθ) cosτ(σ−σ/prime)dτ, (6. 126) where the function fis to be determined. The total potential vanishes on the grounded conical surface, ψ0(σ,α) +ψ1(σ,α) = 0, σ∈(−∞,∞), (6. 127) so that f(τ) =Pm −1 2+iτ(cosα) Pm −1 2+iτ(−cosα). (6. 128) The capacitance Cof the spindle is deduced from the value of the induced potential at the point of inversion A, C=R2ψ1(σ/prime,π) =R/integraldisplay∞ 0sech (πτ)P−1 2+iτ(cosα) P−1 2+iτ(−cosα)dτ. (6. 129) Whenα=1 2π,the spindle degenerates to a sphere of radius a=1 2R,and the value of the capacitance given by (6. 129) coincides with the well-known capacitance C0of the sphere: C0=a.It will be convenient to normalise the capacitance given by (6. 151) against C0.Whenα/lessmuch1, Pm −1 2+iτ(cosα)/similarequal1, Pm −1 2+iτ(−cosα)/similarequal2 πcosh(πτ) ln/parenleftbigg2 α/parenrightbigg , (6. 130) so that the normalised capacitance is approximately C/C 0/similarequal1/ln/parenleftbigg2 α/parenrightbigg , α/lessmuch1. (6. 131) It should be noted that the calculations above are valid when 0 <α≤1 2π. Whenα >1 2π,the image of the spindle under inversion is a spherical shell with circular apertures centred at its poles. ©200 1 CRC Press LLC Figur e6.4 Thespindl ewithvariou sapertures .Theconica lstructure sthatare theirimage sunde rinversio narealsoshown.(a)Anaxisymmetric circula rhole,(b)apairofaxisymmetri ccircula rholes ,and(c)a nonsymmetri cazimuthalslot. Somestructure sforme dbyremovingpartofthespindl esurfac eareshown inFigur e6.4.Unde rinversionthespindl ewithasymmetricall yplace dcir- cularapertur eisequivalenttothesemi-inﬁnit efrustrum ,whils tthespindle withtwosymmetricall yplace dcircula raperture sisequivalenttotheﬁnite conica lfrustrum .Perhap sthemostinterestin gopenspindle-sha pedconduc- torisobtaine dbyintroducin ganazimuthalslot.Unde rinversionitsimageis thesemi-inﬁnit econewithanazimuthalsecto rremoved.Theintroductio nof thisapertur ebreak stheaxialsymmetr yprese ntinalltheconica lstructures considere dabove.Togethe rwiththetoroidwithazimuthalcutsanalyse din Chapte r5,thisstructur eallowsustoillustrat eaverysigniﬁca ntextension of analytic and semi-analytic techniques to the determination of the three di- mensional potential distribution surrounding nonsymmetric open conducting surfaces. ©200 1 CRC Press LLC Consider, therefore, the problem of determining the electrostatic ﬁeld sur- rounding a charged hollow spindle with an azimuthal slot. The equivalent problem is to ﬁnd the electrostatic ﬁeld induced on the grounded semi-inﬁnitecone with an azimuthal (or sectoral) slot by a unit negative charge, located at the inversion centre. Let 2 ϕ 0be the angular width of the sectoral slot. The free-space potential ψ0is given by (6. 125). Based on previous results, the induced potential ψ1may be represented as ψ1(σ,θ,ϕ ) =e−1 2(σ+σ/prime)/integraldisplay∞ 0Fτ(θ,ϕ) cosτ(σ−σ/prime)dτ, (6. 132) where Fτ(θ,ϕ) = sech(πτ)∞/summationdisplay m=0(2−δ0m)fm(τ) cos(mϕ)H(τ,θ), (6. 133) with H(τ,θ) =/braceleftBigg Pm −1 2+iτ(cosθ), θ<θ 0, Pm −1 2+iτ(cosθ0)Pm −1 2+iτ(−cosθ)/Pm −1 2+iτ(−cosθ0), θ>θ 0, and the functions fm(m= 0,1,2,...) are unknowns to be found. The free- space potential may also be written in the analogous form ψ0(σ,θ) =e−1 2(σ+σ/prime)/integraldisplay∞ 0F0 τ(θ) cosτ(σ−σ/prime)dτ, (6. 134) where F0 τ(θ) =−sech(πτ)P−1 2+iτ(cosθ). Forallσ∈(−∞,∞),the following boundary conditions apply to the total potentialψ=ψ0+ψ1, ∂ ∂θψ(σ,θ0−0,ϕ) =∂ ∂θψ(σ,θ1+ 0,ϕ), ϕ∈(0,ϕ0), (6. 135) ψ(σ,θ0−0,ϕ) =ψ(σ,θ1+ 0,ϕ) = 0, ϕ∈(ϕ0,π). (6. 136) Because these boundary conditions apply for the complete interval ( −∞,∞), we may apply a Fourier transform to express them in terms of Fτand its derivative, ∂ ∂θFτ(θ0−0,ϕ) =∂ ∂θFτ(θ0+ 0,ϕ), ϕ∈(0,ϕ0), (6. 137) Fτ(θ0−0,ϕ) =Fτ(θ0+ 0,ϕ) =−F0 τ(θ0),ϕ∈(ϕ0,π).(6. 138) Enforcement of these conditions produces the following dual series equations ∞/summationdisplay m=0(−1)m(2−δ0m) Pm −1 2+iτ(−cosθ0)fm(τ)Γ/parenleftbig1 2+iτ+m/parenrightbig Γ/parenleftbig1 2+iτ−m/parenrightbigcos(mϕ) = 0, ϕ∈(0,ϕ0),(6. 139) ©200 1 CRC Press LLC ∞/summationdisplay m=0(2−δ0m)fm(τ)Pm −1 2+iτ(cosθ0) cos(mϕ) =P−1 2+iτ(cosθ0), ϕ∈(ϕ0,π),(6. 140) where the value of the Wronskian WofPm −1 2+iτ(x) andPm −1 2+iτ(−x) has been employed. We introduce the functions Fm(τ) =(−1)m mΓ/parenleftbig1 2+iτ+m/parenrightbig Γ/parenleftbig1 2+iτ−m/parenrightbigfm(τ) Pm −1 2+iτ(−cosθ0), (6. 141) and separate in (6. 139) and (6. 140) the terms with index m= 0 to obtain ∞/summationdisplay m=1mFm(τ) cos(mϕ) =−f0(τ) 2P−1 2+iτ(−cosθ0), ϕ∈(0,ϕ0), (6. 142) ∞/summationdisplay m=1Gm(τ,θ0)Fm(τ) cos(mϕ) =1 2[1−f0(τ)]P−1 2+iτ(cosθ0), ϕ∈(ϕ0,π), (6. 143) where Gm(τ,θ0) = (−1)mmΓ/parenleftbig1 2+iτ−m/parenrightbig Γ/parenleftbig1 2+iτ+m/parenrightbigPm −1 2+iτ(cosθ0)Pm −1 2+iτ(−cosθ0). (6. 144) We now investigate the asymptotic behaviour of the function Gm(τ,θ0) as m→ ∞.For these purposes τis ﬁxed. From the deﬁnition of the associated Legendre functions (B. 102), Gm(τ,θ0) =mcosh(πτ) πΓ/parenleftbig1 2+iτ+m/parenrightbig Γ/parenleftbig1 2−iτ+m/parenrightbig Γ2(m+ 1)× 2F1(1 2−iτ,1 2+iτ;m+ 1; sin2θ0 2)× 2F1(1 2−iτ,1 2+iτ;m+ 1; cos2θ0 2).(6. 145) Rearrange the Gamma function factors as Γ/parenleftbig1 2+iτ+m/parenrightbig Γ/parenleftbig1 2−iτ+m/parenrightbig Γ2(m+ 1) =Γ2/parenleftbig m+1 2/parenrightbig Γ2(m+ 1)/vextendsingle/vextendsingleΓ/parenleftbig1 2+iτ+m/parenrightbig/vextendsingle/vextendsingle2 Γ2/parenleftbig m+1 2/parenrightbig =Γ2/parenleftbig m+1 2/parenrightbig Γ2(m+ 1)∞/productdisplay n=0/bracketleftBigg 1 +τ2 /parenleftbig n+m+1 2/parenrightbig2/bracketrightBigg−1 .(6. 146) ©200 1 CRC Press LLC FromField’ sformula(seeAppendix ,(B.7))wemaydeduc ethat,asm→∞, Γ2/parenleftbig m+1 2/parenrightbig Γ2(m+1)=1 m/parenleftbigg 1−1 4m+O(m−2)/parenrightbigg . (6.147) More over,itiseasytomaketheestimate ∞/productdisplay n=0/bracketleftBigg 1+τ2 /parenleftbig n+m+1 2/parenrightbig2/bracketrightBigg−1 =1−τ2∞/summationdisplay n=01 /parenleftbig n+m+1 2/parenrightbig2+O(m−2) =1−τ2 m+O(m−2),(6.148) asm→∞.Finall y,fromthedeﬁnitio noftheGaussia nhypergeometri cseries itiseasilyveriﬁedthattheproductofthehypergeometri cfactor soccurring in(6.145)is 1+/parenleftbigg1 4+τ2/parenrightbigg m−1+O(m−2) (6.149) asm→∞.Combinin gtheseestimate sshowsthat Gm(τ,θ0)=π−1cosh(πτ)/parenleftbig 1+O(m−2)/parenrightbig , (6.150) asm→∞. Wetherefor eintroducetheparameter εm(τ)=1−πsech(πτ)Gm(τ,θ0)=O(m−2), (6.151) andrewrit ethedualseries(6.142)and(6.143)intheform ∞/summationdisplay m=1mFm(τ)cos(mϕ)=−f0(τ) 2P−1 2+iτ(−cosθ0),ϕ∈(0,ϕ0), (6.152) ∞/summationdisplay m=1Fm(τ)cos(mϕ)=π 2sech(πτ)[1−f0(τ)]P−1 2+iτ(cosθ0) +∞/summationdisplay m=1εm(τ)Fm(τ)cos(mϕ),ϕ∈(ϕ0,π). (6.153) Whentheslotinthespindl ecloses(ϕ0→0),itmaybeveriﬁedthat f0(τ)=1,fm(τ)=0(m>0) andthesolutio nreduce stothatwhichwaspreviousl yobtaine d(see(6.126) and(6.128)) .Itisclearthatthedualseries(6.152)and(6.153)may besolvedbythestandar dtechniqu efortrigonometri ckernelsoutline din Sectio n2.2.Itisconvenientlydonebysubstitutin gϕ=π−ϑandreplacing ©200 1 CRC Press LLC Fm(τ)=(−1)mF∗ m(τ).Thenϑ0=π−ϕ0istheangularhalf-widthofthe conductorsurfac e(ratherthantheslot). Thesolutionmaynowbededucedfromth edualserie s(2.39),(2.40)and theirsolution(2.61),(2.62)withth efollowingide ntiﬁcationofvalues: m=n,F∗ m(τ)=xn,f0(τ)=x0,qn=εn(τ); a=/braceleftBig 2P−1 2+iτ(−cosθ0)/bracerightBig−1 ,b=π 2sech(πτ)P−1 2+iτ(cosθ0); g0=π 2sech(πτ)P−1 2+iτ(cosθ0);(6.154) theremainin gparameter s(gn,rn,fn,f0)allvanish. Thecapacitanceofth eslotte dspindlemayn owbededuced .Accordingto Bouwkamp’stheorem,itis C=e2σ/primeψ(σ/prime,π,0)=R/integraldisplay∞ 0P−1 2+iτ(cosθ0) cosh(πτ)P−1 2+iτ(−cosθ0)f0(τ)dτ. (6.155) Somefurtherdetailsabou tthecalculationofthisvalu eareprovidedinthe nextsectionwher etheslotte dcharge dspher eisconsidered. 6.4Asphericalshel lwithanazi muthalslot Asremarkedinthepreviou ssection,whe nθ0=1 2π,theslotte dspindle degeneratestoasphericalshellwithanazi muthalslot.Theimageunderthe inversiondescribe dinthatsectionisnotaconewithasectoralslotbu tis, moresimply,aplanewithasectoralcutofhalf-width ϕ0.(SeeFigure6.5.)A caseofparticularinteres tisth ehemisphericalshel landitsimage,thehalf- plane(occurring when ϕ0=1 2π).Th ecapacitanceofthehemispherewas computedinSection 1.4 to be C=a/parenleftbigg1 2+1 π/parenrightbigg . Itprovidesaben chmar kvalu eforsphericalshellswithsectoral slots of arbi- trary angle. When θ0=1 2π,the parameter εm(τ) introduced in (6. 151) may be written in the form εm(τ) = 1−1 2m/vextendsingle/vextendsingleΓ/parenleftbig1 4+1 2iτ+1 2m/parenrightbig/vextendsingle/vextendsingle2 /vextendsingle/vextendsingleΓ/parenleftbig3 4+1 2iτ+1 2m/parenrightbig/vextendsingle/vextendsingle2. (6. 156) Although the parameter has a simpler form than when 0 < θ 0<1 2π,it is still not possible to solve the associated potential problem in a closed form. ©200 1 CRC Press LLC Figur e6.5 Aspherica lshellwithanazimuthalslot;itsimag eunde rinversion isthexOyplanewithasectora lslotremoved. However,itispossibl etoobtai nsomeanalytica lapproximation sintwolim- itingcases :thenarrowcut(ϕ0/lessmuch1)andthenarrowsectora lconductor (ϑ0=π−ϕ0/lessmuch1).Theproble mhassomesimilaritie swiththeazimuthally slotte ddegenerat etorustreate dinChapte r5,andsosomerepetitiou sdetails will be suppressed. Settingθ0=1 2π,it follows from (6. 155) that the capacitance for a spherical shell with an azimuthal slot is C=R/integraldisplay∞ 0sech(πτ)f0(τ)dτ. (6. 157) ©200 1 CRC Press LLC Making use of the identiﬁcation (6. 154), we may recognise that f0(τ) =/braceleftbigg b(τ)−a(τ) ln/bracketleftbigg1−t0 2/bracketrightbigg/bracerightbigg−1 × /braceleftBigg b(τ) +1 +t0 2∞/summationdisplay n=1/parenleftbigg2 n/parenrightbigg1 2 F∗ n(τ)εn(τ)ˆP(0,1) n−1(t0)/bracerightBigg (6. 158) where the parametric dependence of a=a(τ) andb=b(τ) in (6. 154) is made explicit; t0= cosϑ0. Whenϕ0=π−ϑ0/lessmuch1,it is readily observed from (6. 158) that f0(τ) =b(τ)/braceleftBig b(τ)−2a(τ) ln cosϕ0 2/bracerightBig−1 +O(ϕ2 0). (6. 159) When the spindle closes, the function f0(τ) becomes 1; thus if we deﬁne ε=−ln cosϕ0 2, then f0(τ) = 1−2a(τ)ε {b(τ) + 2a(τ)ε}+O(ϕ2 0) = 1−2a(τ) b(τ)ε{1 +O(ε)}(6. 160) when the slot is logarithmically narrow (ε/lessmuch1). When the conductor is a narrow sector ( ϑ0/lessmuch1), we may deduce from (6. 158) that f0(τ) =/braceleftbigg b(τ)−2a(τ) ln sinϑ0 2/bracerightbigg−1/braceleftBigg b(τ) +∞/summationdisplay n=1F∗ n(τ)εn(τ) +O(ϑ2 0)/bracerightBigg , (6. 161) where the functions F∗ nmay be approximated as the solution of the inﬁnite system (with a conﬂuent matrix) F∗ m(τ)−2a(τ) b(τ)−2a(τ) ln sin1 2ϑ01 m∞/summationdisplay n=1F∗ n(τ)εn(τ) =1 m2a(τ)b(τ) b(τ)−2a(τ) ln sin1 2ϑ0.(6. 162) This system can be solved by multiplying both sides of (6. 162) by εm(τ) and summing over m.Thus ∞/summationdisplay n=1F∗ n(τ)εn(τ) =2a(τ)b(τ)A(τ) b(τ)−2a(τ)/braceleftbig A(τ) + ln sin1 2ϑ0/bracerightbig, (6. 163) ©200 1 CRC Press LLC where A(τ) =∞/summationdisplay m=1εm(τ) m. (6. 164) Insertion of (6. 163) in (6. 161) shows that f0(τ) =b(τ) b(τ)−2a(τ)/braceleftbig A(τ) + ln sin1 2ϑ0/bracerightbig/braceleftbig 1 +O(ϑ2 0)/bracerightbig . (6. 165) If we introduce the parameter ε/prime=−/braceleftbigg ln sin1 2ϑ0/bracerightbigg−1 , then f0(τ) =b(τ) 2a(τ)ε/prime{1 +O(ε/prime)}, (6. 166) when the sector is logarithmically narrow ( ε/prime/lessmuch1). Thus the capacitance of the logarithmically narrow slot is C1=C0−2 πεR/integraldisplay∞ 0/braceleftBig P−1 2+iτ(−cosθ0)/bracerightBig−2 dτ+O(ε2), (6. 167) whereC0is the capacitance of the corresponding closed spindle (see (6. 129)), and the capacitance of the logarithmically narrow sector is C2=π 2ε/primeR/integraldisplay∞ 0sech2(πτ)/braceleftBig P−1 2+iτ(cosθ0)/bracerightBig2 dτ+O(ε/prime2). (6. 168) Whenθ0=1 2π,tabulated values of the integrals occurring in (6. 167) and (6. 168) are1 16π2and1 2,respectively (see [15]), so that C1/C0= 1−π 4ε+O(ε2), C2/C0=π 2ε/prime+O(ε/prime2). (6. 169) Now consider the needle-shaped spindle ( θ0/lessmuch1) with a logarithmically narrow slot ( ε/lessmuch1) and, in addition, suppose that ϕ0/lessmuchθ0.The approxima- tion for the capacitance is C1 C0/similarequal1 ln(2/θ0)/braceleftbigg 1−ε ln(2/θ0)/bracerightbigg . (6. 170) Whenθ0/lessmuch1 andϑ0/lessmuch1,the structure very nearly becomes a straight ﬁnite strip with some variable width and its capacitance is approximately C2 C0/similarequalε/prime. (6. 171) Comparing (6. 169) and (6. 171), we may recognise the diﬀerence of a factor of1 2πin capacitance between the spherically curved crescent-shaped strip (6. ©200 1 CRC Press LLC 169) and its ﬂat analogy (6. 171). On the other hand, the characteristic factor of {ln(2/θ0)}−1present in (6. 170) is notably absent in (6. 171). It is therefore important to recognise that these approximations are not uniformly valid in the problem parameters, and that the regime of their validity is best delineated by numerical methods; nonetheless, the approximations are useful at the extreme limit of the parameter range. ©200 1 CRC Press LLC Chapter 7 Two-dimensional Potential Theory Historically, two-dimensional potential problems have been studied more ex- tensively than have three-dimensional problems. Apart from the apparentsimplicity of lower dimension, the main reasons are that powerful methods,based upon conformal mapping techniques and the well-developed theory ofanalytic functions, are available in the plane; these provide rather clear proce- dures to facilitate the solution of mixed boundary value problems in potential theory. Basically, analytic function theory techniques reduce the potential problem to the well-known Riemann-Hilbert problem of the determination of an an-alytical function on some contour bounding a domain [45]; various concreteapplications of this technique can be found in [18] and [53]. Applications of the conformal mapping method are so numerous that classic texts on electromag- netic theory invariably describe and solve a variety of electrostatic problems with this technique (see, for example, [54, 66]). Despite the lower dimension, it should be observed that boundary value problems in two-dimensional potential theory involve an additional abstrac-tion compared to that for three-dimensional bodies of ﬁnite extent, even for open surfaces with sharp edges. Whilst it is reasonable to imagine an ex- tremely long, but at the same time ﬁnite conductor charged to some poten-tial, its extension to inﬁnity, at the same constant potential as for the ﬁniteconductor, raises some questions about the physical reality or relevance of themodel. A physicist might reasonably question the source of inﬁnite energy needed to charge this inﬁnitely long conductor. It is not surprising, then, that two-dimensional potential problems, even properly stated, require some nonphysical behaviour of the potential function at inﬁnity. This manifests itself as a logarithmic dependence on distance from the conductors, so the potential is unbounded at inﬁnity. Although strangefrom the physical perspective, the mathematical issue simply concerns the choice of the class of functions required for a satisfactory two-dimensional potential theory. Generally speaking, if the conductor is modelled as an in-ﬁnitely long object of constant cross-section, the basic postulates of potentialtheory force a logarithmic increase to solutions at large distances from theconductor. Some simple illustrative examples will indicate distinctive features of two- dimensional potentials. The electrostatic potential ψ(− →r), due to some elec- ©200 1 CRC Press LLC triﬁedconductorhel datunitpote ntialintwodimensions ,isdeﬁnedbythe single-laye rpotential[66] ψ(−→r)=−1 2π/integraldisplay Llog/vextendsingle/vextendsingle/vextendsingle−→r−−→ r/prime/vextendsingle/vextendsingle/vextendsingleσ/parenleftBig−→ r/prime/parenrightBig dl (7.1) whereG2(−→r,−→r/prime)=−(1/2π)log|−→r−−→r/prime|istwo-dimensionalGreen’ sfunc- tion,σisthelinearchargedensityonthecross-sectionalcontou rL,dlisthe elementofthecontou rintegral,and−→r,−→r/primearepositionvectorsofobservation pointsandpointson L,respecti vely. Wewillconside ravarietyofcanonicalstructuresthatar einﬁnitecylin- dersofconstantcross-section ,intowhichaperture sarei ntroducedtoproduce longitudinallyslottedcylinders(theedge softh eslotsar eparalleltothecylin- dricalaxis). Bywayofintroduction ,weconsiderth ecirculararc(Section7.1),andthen circularcylinderswithmultipleslots(Section7.2),variou sconﬁguration sof thinstrip s(Section7.3) ,andellipticcylinderswit hmultipleslots(Section7.4). InSection7.5 ,asingly-slottedcylinderwitharbitrarycross-sectio nisconsid- ered.Althoughthisstructureisnoncanonical ,ourpurposeistodemonstrate howtoregularis etheintegralequationsofpotentialtheoryinarathermore generalsettingthanthesimple rcanonicalstructuresdiscussedintheearlier sections.Theproces stransformstheintegralequation stoasecond-kin dsys- temofequationswithitsattendantbeneﬁts:awell-conditione dsyste mof equationsfornumericalsolutionaftertruncation. 7.1Thecirculararc Consideraninﬁnitelylong,singly-slotte dcircularcylinderwhosecross- sectionisanarc Lofacircl eofradius a(seeFigure7.1).Polarc oordinates (r,ϕ),wherer=ρ/a, areconvenie ntforthisconﬁguration .Assum ethat the right half of the arc (given by ϕ∈(0,ϕ0)) is charged to unit potential, but the left half (given by ϕ∈(−ϕ0,0)) is charged either to unit positive or negative value, i.e., ψ(1,ϕ) = (−1)l,(l= 0,1).Ifσldenotes the charge distribution on L,it is evident that σl(−ϕ) = (−1)lσl(ϕ).(Whenl= 1,an inﬁnitesimally small insulating gap is placed at ϕ= 0.) Then the potential ©200 1 CRC Press LLC Figure 7.1 The circular arc. can be represented as ψl(r,ϕ) =−1 4πϕ0/integraldisplay −ϕ0log/vextendsingle/vextendsingle1−2rcos(ϕ−ϕ/prime) +r2/vextendsingle/vextendsingleσl(ϕ/prime)dϕ/prime =−1 4πϕ0/integraldisplay 0Kl(r,ϕ,ϕ/prime)σl(ϕ/prime)dϕ/prime, (7. 2) where Kl(r,ϕ,ϕ/prime) = log/vextendsingle/vextendsingle1−2rcos(ϕ−ϕ/prime) +r2/vextendsingle/vextendsingle + (−1)llog/vextendsingle/vextendsingle1−2rcos(ϕ+ϕ/prime) +r2/vextendsingle/vextendsingle.(7. 3) Whenl= 0,it can be readily shown that ψl(r,ϕ)−1 2πqlog/parenleftbig r−1/parenrightbig is a regular harmonic function, as r→ ∞,where q= 2ϕ0/integraldisplay 0σ0(ϕ/prime)dϕ/prime ©200 1 CRC Press LLC/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1x y o−ϕοϕοϕ=0 ϕ=π is the (total) charge per unit length; when l= 1,ψl(r,ϕ) behaves as a regular harmonic function, as r→ ∞. First consider the uniformly charged strip ( l= 0).Using the expansion [19], log(1−2tcos (ϕ−ϕ/prime) +t2) =−2∞/summationdisplay k=1tk kcoskx, (7. 4) the kernel of 7. 2 has the cosine Fourier series K0(r,ϕ,ϕ/prime) =  −4∞/summationtext n=1n−1rncosnϕcosnϕ/prime, r< 1, 4 logr−4∞/summationtext n=1n−1r−ncosnϕcosnϕ/prime, r> 1.(7. 5) Extend the domain of deﬁnition of σ0,deﬁning σ0 tot(ϕ/prime) =/braceleftbiggσ0(ϕ/prime), ϕ/prime∈(0,ϕ0) 0, ϕ/prime∈(ϕ0,π); (7. 6) this even function has a Fourier series expansion σ0 tot=∞/summationdisplay m=0/parenleftbig 2−δ0 m/parenrightbig xmcosmϕ/prime, (7. 7) with unknown Fourier coeﬃcients xmto be determined. Substitute these expansions into (7 .2) to obtain ψ(r,ϕ) =  ∞/summationtext n=1n−1xnrncosnϕ, r< 1 −x0logr+∞/summationtext n=1n−1xnr−ncosnϕ, r> 1. (7. 8) This representation can also be obtained by the method of separation of vari- ables applied directly to Laplace’s equation. The boundary condition at r= 1, ϕ∈(0,ϕ0) is ψ(1 + 0,ϕ) =ψ(1−0,ϕ) = 1; (7. 9) on the slot r= 1,ϕ∈(ϕ0,π),the boundary condition, which follows directly from the deﬁnition (7 .6),is σ0 tot(ϕ) =∂ψ(r,ϕ) ∂r/vextendsingle/vextendsingle/vextendsingle/vextendsingler=1+0 r=1−0= 0. (7. 10) Enforcement of these boundary conditions produces the following dual series equations: ∞/summationdisplay n=11 nxncosnϕ= 1, ϕ ∈(0,ϕ0) (7. 11) ∞/summationdisplay n=1xncosnϕ=−1 2x0, ϕ ∈(ϕ0,π). (7. 12) ©200 1 CRC Press LLC Themethodde velope dinSection2. 2showsthatth eclose dformsolution is x0=−/braceleftBig log/parenleftBig sinϕ0 2/parenrightBig/bracerightBig−1 , xm=−1 2/braceleftBig log/parenleftBig sinϕ0 2/parenrightBig/bracerightBig−1 (1+cosϕ0)P(0,1) m−1(cosϕ0) =−1 2/braceleftBig log/parenleftBig sinϕ0 2/parenrightBig/bracerightBig−1 {Pm(cosϕ0)+Pm−1(cosϕ0)},(7.13) whenm> 0.Thecapacitanceofthecylindricall yshapedstri p(pe runit length)isthus C=2π/integraldisplay 0σ0 tot(ϕ/prime)dϕ/prime=2πx0=−2π/braceleftBig log/parenleftBig sinϕ0 2/parenrightBig/bracerightBig−1 . (7.14) Ontheinterval[0 ,ϕ0],thelin echargedensi tyequals σ0 tot(ϕ)=1 4π∂ψ(r,ϕ) ∂r/vextendsingle/vextendsingle/vextendsingle/vextendsingler=1+0 r=1−0=1 4π∞/summationdisplay n=0/parenleftbig 2−δ0 n/parenrightbig xncosmϕ, (7.15) anditsvalu eiseasilydeduce dfromthedisconti nuou sserie s(1.109)t obe σ0 tot(ϕ)=−1 2√ 2πcosϕ 2/braceleftBig log/parenleftBig sinϕ0 2/parenrightBig/bracerightBig−1 (cosϕ−cosϕ0)−1 2,ϕ<ϕ 0; (7.16) itvanishe swhe nϕ>ϕ 0. Whenth ecirculararccomprisesoppositelycharge dhalve s(Figur e7.1),the potential is bounded; there is no logarithmic term. Physically, the structure is a two-dimensional dipole. Set l= 1 in (7.2) and again use expansion (7 .4) to obtain K1(r,ϕ;ϕ/prime) =−4∞/summationdisplay n=1n−1sinnϕsinnϕ/prime/braceleftbigg rn, r< 1. r−n,r>1.(7.17) As before, introduce the extended or total line charge density σ1 tot(ϕ/prime) =/braceleftbigg σ0(ϕ/prime), ϕ/prime∈(0,ϕ0) 0, ϕ/prime∈(ϕ0,π), (7. 18) and represent this odd function as a Fourier sine series σ1 tot(ϕ/prime) =∞/summationdisplay m=1ymsinmϕ/prime. (7. 19) Substitute (7 .17) and (7.19) into (7.2) to obtain ψ1(r,ϕ) =∞/summationdisplay n=1n−1ynsinnϕ/braceleftbiggrn, r< 1. r−n,r>1.(7.20) ©200 1 CRC Press LLC Enforcingthemixe dboundaryconditionsonthearc r=1produce sthedual seriesequation sfortheunkn owncoeﬃcie ntsyn: ∞/summationdisplay n=1n−1ynsinnϕ=1,ϕ ∈(0,ϕ0)(7.21) ∞/summationdisplay n=1ynsinnϕ=0,ϕ ∈(ϕ0,π). (7.22) Thesolutionofthes eequation s(seeSection2.2)is(with z0=cosϕ0), yn=√ 2 πn/integraldisplay1 z0P(0,1) n−1(t) (1−t)1 2dt=√ 2 πn/integraldisplay1 z0Pn(t)+Pn−1(t) (1+t)(1−t)1 2dt. (7.23) Theformatofthissolution(7 .23)hassomerathersatisfactoryfeatures.For example,onemayconvenientl ycalculateth edistributionofth epote ntialon thecircler=1tobe ψ(1,ϕ)=2π−1arctan/bracketleftbigg√ 2sin1 2ϕ0cos1 2ϕ{cosϕ0−cosϕ}−1 2/bracketrightbigg (7.24) whenϕ>ϕ 0;when0<ϕ<ϕ 0,ψ(1,ϕ)=1. 7.2Axiallyslottedopencircularcylinders Inthissection,slotte dcircularcylinderswithmultipleaperturesar econ- sidered.Arestricte dselectionofelectrostaticproblemsthatar edistincti ve ofthisgeometr yareexamined .Ourﬁrstcalculationi softh eelectrostatic ﬁeldduetoapairofchargedcirculararcs ,asymmetricallyplacedasshownin Figure7.2.Th esecondcalculationisoftheﬁeldgeneratedbyth equadrupole lensalsosh owni nFigure7.2;forth esakeofsimplici ty,whe nthearc sareall positivel ychargedwerestric tatte ntiont othesymmetricalcas e(ϕ1=π−ϕ0). Asintheprevioussection,th econductor slieonthecontouroftheunit circleandarechargedt opote ntialsV1=1andV2=(−1)lasshowninFigure 7.2;theindex l=0or1.Thepotentialassociatedwithth epairofcharged circular arcs, at potentials V1(deﬁned by ϕ∈(ϕ0,ϕ1)) andV2(deﬁned by ϕ∈(−ϕ1,−ϕ0)) is ψl(r,ϕ) =−1 4π/integraldisplayϕ1 ϕ0Kl 1(r,ϕ,ϕ/prime)σl(ϕ/prime)dϕ/prime, (7. 25) whereKl 1=Klis deﬁned by (7. 3). For the two-dimensional quadrupole lens in which the pair of arcs deﬁned by ϕ∈(ϕ0,ϕ1)∪(−(π−ϕ0),−(π−ϕ1)) ©200 1 CRC Press LLC Figure 7.2 The circular arc (left) and quadrupole (right). is held at potential V1, and the pair of arcs deﬁned by ϕ∈(−ϕ0,−ϕ1)∪ (π−ϕ0,π−ϕ1) is held at potential V2,the potential is ψl(r,ϕ) =−1 4π/integraldisplayϕ1 ϕ0K2(r,ϕ,ϕ/prime)σl(ϕ/prime)dϕ/prime(7. 26) where K2(r,ϕ,ϕ/prime) = log/bracketleftBig/parenleftbig r2+ 1/parenrightbig2−4r2cos2(ϕ−ϕ/prime)/bracketrightBig + (−1)llog/bracketleftBig/parenleftbig r2+ 1/parenrightbig2−4r2cos (ϕ+ϕ/prime)/bracketrightBig . First consider the pair of charged arcs. Enforcement of the boundary con- dition ψl(1 + 0,ϕ) =ψl(1−0,ϕ) = 1, ϕ ∈(ϕ0,ϕ1), (and the corresponding condition on the arc at potential V2) produces a ﬁrst-kind Fredholm integral equation for the unknown charge density σl.On equicharged arcs ( l= 0),the density σ0satisﬁes −1 2π/integraldisplayϕ1 ϕ0σ0(ϕ/prime) ln 4/vextendsingle/vextendsingle/vextendsingle/vextendsinglesin21 2ϕ−sin21 2ϕ/prime/vextendsingle/vextendsingle/vextendsingle/vextendsingledϕ/prime= 1, ϕ ∈(ϕ0,ϕ1),(7. 27) ©200 1 CRC Press 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σ1satisﬁes 1 2π/integraldisplayϕ1 ϕ0σ1(ϕ/prime)log/vextendsingle/vextendsingle/vextendsingle/vextendsingletan1 2ϕ+tan1 2ϕ/prime tan1 2ϕ−tan1 2ϕ/prime/vextendsingle/vextendsingle/vextendsingle/vextendsingledϕ/prime=1,ϕ ∈(ϕ0,ϕ1).(7.28) FollowingtheargumentofSection7.1,thes eintegralequation smaybe replaced by the triple series equations   ∞/summationtext n=1xncosnϕ=−1 2x0, ϕ ∈(0,ϕ0)∪(ϕ1,π), ∞/summationtext n=1n−1xncosnϕ= 1, ϕ ∈(ϕ0,ϕ1),(7. 29) and  ∞/summationtext n=1ynsinnϕ= 0, ϕ ∈(0,ϕ0)∪(ϕ1,π), ∞/summationtext n=1n−1ynsinnϕ= 1, ϕ ∈(ϕ0,ϕ1),(7. 30) where the densities σ0andσ1are respectively expanded in cosine and sine Fourier series, σ0(ϕ/prime) =∞/summationdisplay m=0/parenleftbig 2−δ0 m/parenrightbig xmcosmϕ/prime, (7. 31) σ1(ϕ/prime) =∞/summationdisplay n=1ynsinnϕ/prime, (7. 32) with unknown coeﬃcients {xn}∞ n=0and{yn}∞n=1. The symmetric situation ( ϕ1=π−ϕ0) is quickly solved. The odd index coeﬃcients all vanish, and the Equations (7 .29) reduce to the following dual series equations for the even index coeﬃcients x2n, ∞/summationtext n=1x2ncosnϑ=−1 2x0, ϑ ∈(0,ϑ0), ∞/summationtext n=1n−1x2ncosnϑ= 2, ϑ ∈(ϑ0,π),(7. 33) whereϑ= 2ϕandϑ0= 2ϕ0. The substitution ϑ=π−θtransforms (7. 33) to∞/summationtext n=1n−1Xncosnθ= 2, θ ∈(0,θ0), ∞/summationtext n=1Xncosnθ=−1 2x0, θ ∈(θ0,π),(7. 34) whereXn= (−1)nx2n, θ0=π−ϑ0=π−2ϕ0,andθ=π−2ϕ.Comparing Equations (7 .11) and (7.12) with (7 .34), the solution of this symmetric case is x0=−2{log (cosϕ0)}−1, x2n=−{log (cosϕ0)}−1{Pn(cosϕ0)−Pn−1(cosϕ0)}.(7. 35) ©200 1 CRC Press LLC Nowconsider(7 .30)inthemoregeneralcase ,inwhichtheparameter sϕ0 andϕ1areunrelated,takingarbitrar yvalue sin(0,π),withϕ0<ϕ 1.The standard formoftripleseriesequationsinvolvin gthetrigonometricalkernels {sinnϕ}∞ n=1isobtainedbysettin gyn=nan;Equation s(7.30)become ∞/summationtext n=1nansinnϕ=0,ϕ ∈(0,ϕ0)∪(ϕ1,π), ∞/summationtext n=1ansinnϕ=1,ϕ ∈(ϕ0,ϕ1).(7.36) FromtheresultsofSection2.7,Equations(7 .36)ar eequi valenttothesym- metric triple equations ∞/summationtext n=1nbnsinnϑ= 0, ϑ ∈(0,ϑ0)∪(π−ϑ0,π), ∞/summationtext n=1bnsinnϑ=/parenleftbig tan1 2ϕ0tan1 2ϕ1/parenrightbig1 2, ϑ ∈(ϑ0,π−ϑ0),(7. 37) where tan1 2ϑ0= tan1 2ϕ0cot1 2ϕ1. In turn these equations may be reduced to the following dual series equa- tions for the odd index Fourier coeﬃcients b2n+1, ∞/summationtext n=0/parenleftbig n+1 2/parenrightbig b2n+1sin/parenleftbig n+1 2/parenrightbig θ= 0, θ ∈(0,θ0) ∞/summationtext n=0b2n+1sin/parenleftbig n+1 2/parenrightbig θ=/parenleftbig tan1 2ϕ0tan1 2ϕ1/parenrightbig1 2, θ ∈(θ0,π)(7. 38) whereθ= 2ϑ,andθ0= 2ϑ0; all the even index coeﬃcients b2nvanish. It should be noted that original coeﬃcients {an}∞n=1are related to {bn}∞n=1 by (2.263).By means of the Abel integral transform, we deduce from Equa- tions (7.38) that ∞/summationdisplay n=0b2n+1Pn(z) =2 π/parenleftBig tanϕ0 2tanϕ1 2/parenrightBig1 2/braceleftbigg F1(z), z ∈(−1,z0), F2(z), z ∈(z0,1),(7. 39) wherez= cosθandz0= cosθ0,and F1(z) =K/parenleftBigg/radicalbigg 1 +z 2/parenrightBigg , F2(z) =K/parenleftBigg/radicalbigg 1−z 2/parenrightBigg K/parenleftBigg/radicalbigg 1 +z0 2/parenrightBigg /K/parenleftBigg/radicalbigg 1−z0 2/parenrightBigg . (Kis the complete elliptic integral of the ﬁrst kind.) Orthogonality of the ©200 1 CRC Press LLC Legendrepolynomial son[−1,1]instantlyimplies b2n+1=2 π/parenleftBig tanϕ0 2tanϕ1 2/parenrightBig1 2/parenleftbigg n+1 2/parenrightbigg ×   z0/integraldisplay −1F1(z)Pn(z)dz+1/integraldisplay z0F2(z)Pn(z)dz  .(7.40) Theintegralsoccurrin gin(7.40)ar ereadilycalculated,ifon erecall sthe relationshipbetweenth ecompleteellipticintegralsandtheLegendrefunctions P−1 2(z)=2 πK/parenleftBigg/radicalbigg 1−z 2/parenrightBigg ,Q−1 2(z)=K/parenleftBigg/radicalbigg 1+z 2/parenrightBigg . (7.41) Anintegrationbypartsandus eofthediﬀere ntialequationfortheLegendre functionsproduce sthecompactresult b2n+1=/parenleftBig tanϕ0 2tanϕ1 2/parenrightBig1 2/braceleftBigg/parenleftbigg n+1 2/parenrightbigg K/parenleftBigg/radicalbigg 1−z0 2/parenrightBigg/bracerightBigg−1 Pn(z0).(7.42) Thecapacitanceoftheseoppositelychargedcirculararcsequals C=1 4π∞/summationdisplay n=0a2n+1=1 4π/parenleftBig tanϕ0 2tanϕ1 2/parenrightBig1 2∞/summationdisplay n=0b2n+1 =1 2πK/parenleftBigg/radicalbigg 1+z0 2/parenrightBigg /K/parenleftBigg/radicalbigg 1−z0 2/parenrightBigg .(7.43) Intermsofth eoriginalparametersthiscapacitanceis C=1 2πK/parenleftbigg1−q 1+q/parenrightbigg /K/parenleftbigg2√q 1+q/parenrightbigg =1 2πK(t)/K/parenleftBig/radicalbig 1−t2/parenrightBig , (7.44) whereq=tan1 2ϕ0cot1 2ϕ1,andt=sin1 2(ϕ1−ϕ0)/sin1 2(ϕ1+ϕ0). Thecalculationofth elinechargedensi tyσbasedontheevidentrelationship betweenth etransforme dseries(se eSection2.7)is σ(ϕ) =1 4π∞/summationdisplay n=1nansinnϕ =1 4π/parenleftBig sin2ϕ 2+ tanϕ0 2tanϕ1 2cos2ϕ 2/parenrightBig−1∞/summationdisplay n=1nbnsinnϑ (7. 45) where tan1 2ϑ= tanϕ 2/braceleftBig tanϕ0 2tanϕ1 2/bracerightBig−1 2. ©200 1 CRC Press LLC Suppressing the details of an uncomplicated but bulky transformation, we deduce the ﬁnal formula for the charge density on the asymmetric disposedarcs to be σ(ϕ) =1 K/parenleftbig√ 1−t2/parenrightbigsin1 2(ϕ1+ϕ0)/radicalbig (cosϕ0−cosϕ) (cosϕ−cosϕ1)(7. 46) where the parameter twas deﬁned above. Finally, consider the quadrupole lens charged so that the potentials V1= −V2= 1 (l= 1).From (7.26) the electrostatic potential is ψ1(r,ϕ) =−1 4π/integraldisplayϕ1 ϕ0log/bracketleftbigg1−2r2cos (2ϕ−2ϕ/prime) +r4 1−2r2cos (2ϕ+ 2ϕ/prime) +r4/bracketrightbigg σ1(ϕ/prime)dϕ/prime.(7. 47) We expand the kernel of Equation (7. 47) as −4∞/summationdisplay n=1n−1sin 2nϕsin 2nϕ/prime/braceleftbiggr2n, r< 1 r−2n,r>1(7.48) and line charge density as a Fourier sine series σ(ϕ/prime) =∞/summationdisplay n=1y2nsin 2nϕ/prime, (7. 49) so that (cf. (7 .20)) ψ1(r,ϕ) =∞/summationdisplay n=1n−1y2nsin 2nϕ/braceleftbiggr2n, r< 1 r−2n,r>1.(7.50) By the same argument as above, we obtain triple series equations for the coeﬃcients {y2n}∞ n=1, ∞/summationtext n=1y2nsin 2nφ= 0, φ ∈(0,φ0)∪(φ1,π), ∞/summationtext n=1n−1y2nsin 2nφ= 1, φ ∈(φ0,φ1),(7. 51) whereφ= 2ϕ, φ 0= 2ϕ0,andφ1= 2ϕ1. Equations (7 .30) and (7.51) are the same, so that the solution of (7 .51) is given by (7.42) with replacement of the parameters φ0andφ1by 2ϕ0and 2ϕ1 respectively. With this replacement, Formulae (7 .44) and (7.46) hold for the quadrupole lens. In principle, more complicated conﬁgurations of cylindrical strips lying on the contour of a circle may be tackled by this approach. The resulting series equations are naturally more complex, but considerable simpliﬁcation occursif the components are symmetrically located. ©200 1 CRC Press LLC Figur e7.3 Pairsofcharge dthinstrips. 7.3Electrostati cpotentialofsystem sofcharge dthin strips Inmanyrespects,potentialproblem sforﬂatstripsaresimila rtothosefor cylindrically-sha pedstrips .Anotabl ediﬀerenc eistheextractio nofzeroterms inthefunctiona lequation swithcontinuousspectru m(integra lequations), whichisanalogou stotheextractio nofzero-orde rFourie rcoeﬃcie ntsinseries equations. Letusconside rthecanonica lexampl eofthepairofcharge dcoplana rﬂat stripsshowninFigur e7.3;thestripsoccupytheregion sa≤y/prime≤b,−b≤ y/prime≤ −aand are charged to potentials V1= 1,V2(−1)lrespectively, where l= 0 or 1. It is convenient to solve this problem in rescaled Cartesian coordinates ( ρ,z) derived from standard coordinates ( y/prime,z/prime) byρ=y/prime/(ab)1 2;z=z/prime/(ab)1 2. Thus the strips occupy the regions ρ0≤ρ≤ρ1,−ρ1≤ρ≤ −ρ0, where ρ0= (a/b)1 2andρ1=ρ−1 0= (b/a)1 2. The total potential ψis the sum of single-layer potentials ψ1,ψ2derived from the right-half plane and left-half plane strips, respectively: ψ(ρ,z) =ψ1(ρ,z) +ψ2(ρ,z), (7. 52) where ψ1(ρ,z) =−1 4π/integraldisplayρ1 ρ0ln/bracketleftBig (ρ−ρ/prime)2+z2/bracketrightBig σ1(ρ/prime)dρ/prime, (7. 53) ψ2(ρ,z) =−1 4π/integraldisplay−ρ0 −ρ1ln/bracketleftBig (ρ−ρ/prime)2+z2/bracketrightBig σ2(ρ/prime)dρ/prime. (7. 54) ©200 1 CRC Press LLC−ρ −ρ ρ ρ ab -a -bVV oo1 2 1V V2 1001z z a) b) yy The symmetry of the problem instantly implies σ1(ρ/prime) =σ2(−ρ/prime)def=σ0(ρ/prime) ifl= 0, σ1(ρ/prime) =−σ2(−ρ/prime)def=σ1(ρ/prime) ifl= 1, so that a single representation for the potential is ψl(ρ,z) =−1 4π/integraldisplayρ1 ρ0K(ρ,z;ρ/prime,z/prime)σl(ρ/prime)dρ/prime(7. 55) where the kernel K(ρ,z;ρ/prime) = log/bracketleftbigg/radicalBig (ρ−ρ/prime)2+z2/bracketrightbigg + (−1)llog/bracketleftbigg/radicalBig (ρ+ρ/prime)2+z2/bracketrightbigg . A ﬁrst-kind Fredholm integral equation for the line charge density σlis obtained by enforcement of the boundary condition on the strips, ψl(ρ,+0) =ψl(ρ,−0) = 1, ρ ∈(ρ0,ρ1), (7. 56) yielding −1 2π/integraldisplayρ1 ρ0σ0(ρ/prime) log/vextendsingle/vextendsingleρ2−ρ/prime2/vextendsingle/vextendsingledρ/prime= 1, ρ ∈(ρ0,ρ1), (7. 57) and 1 2π/integraldisplayρ1 ρ0σ1(ρ/prime) log/vextendsingle/vextendsingle/vextendsingle/vextendsingleρ+ρ/prime ρ−ρ/prime/vextendsingle/vextendsingle/vextendsingle/vextendsingledρ/prime= 1, ρ ∈(ρ0,ρ1). (7. 58) We now use familiar mathematical tools to reduce both equations to triple integral equations for some unknown Fourier coeﬃcients. First, represent the logarithmic kernels by their Fourier transforms log/vextendsingle/vextendsingleρ2−ρ/prime2/vextendsingle/vextendsingle= 2 logρ+ log/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−ρ /prime2 ρ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle = 2 logρ+ 2/integraldisplay ∞ 01−cosξρ/prime ξcosξρdξ, (7. 59) and log/vextendsingle/vextendsingle/vextendsingle/vextendsingleρ+ρ/prime ρ−ρ/prime/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 2/integraldisplay ∞ 0sinξρ/prime ξsinξρdξ. (7. 60) Then extend the domain of σlby introducing the functions σl totwith their associated Fourier transforms, σ0 tot=/braceleftbigg0, ρ ∈(0,ρ0)∪(ρ1,∞) σ0(ρ), ρ ∈(ρ0,ρ1)/bracerightbigg =/integraldisplay∞ 0g(λ) cos (λρ)dλ, (7. 61) ©200 1 CRC Press LLC σ1 tot=/braceleftbigg 0, ρ ∈(0,ρ0)∪(ρ1,∞) σ1(ρ), ρ ∈(ρ0,ρ1)/bracerightbigg =/integraldisplay∞ 0f(λ) sin (λρ)dλ. (7. 62) Equations (7 .57) and (7.58) can be replaced by the equivalent integral equa- tions on an extended range of integration, −1 2π/integraldisplay∞ 0σ0 tot(ρ/prime) log/vextendsingle/vextendsingleρ2−ρ/prime2/vextendsingle/vextendsingledρ/prime= 1, ρ ∈(ρ0,ρ1) (7. 63) and 1 2π/integraldisplay∞ 0σ1 tot(ρ/prime) log/vextendsingle/vextendsingle/vextendsingle/vextendsingleρ+ρ/prime ρ−ρ/prime/vextendsingle/vextendsingle/vextendsingle/vextendsingledρ/prime= 1, ρ ∈(ρ0,ρ1). (7. 64) Substitution of the Fourier transforms for the kernels (7 .59) and (7.60), and unknown functions σl((7.61) and(7.62)), produces the integral equations /integraldisplay∞ 0ξ−1[g(ξ)−g(0)] cos (ξρ)dξ= 2 +g(0) logρ, ρ∈(ρ0,ρ1),(7. 65) /integraldisplay∞ 0ξ−1f(ξ) sin (ξρ)dξ= 2, ρ ∈(ρ0,ρ1). (7. 66) Equations (7 .61) and (7.62) provide the complementary part of the triple integral equations for unknown functions gandf, respectively: for oppositely charged ﬂat strips ( l= 1), /integraldisplay∞ 0f∗(ξ) sin (ξρ)dξ= 0, ρ ∈(0,ρ0)∪(ρ1,∞), /integraldisplay∞ 0ξ−1f∗(ξ) sin (ξρ)dξ= 1, ρ ∈(ρ0,ρ1), (7. 67) whereas for positively charged strips ( l= 0), /integraldisplay∞ 0g∗(ξ) cos (ξρ)dξ= 0, ρ ∈(0,ρ0)∪(ρ1,∞), /integraldisplay∞ 0ξ−1[g∗(ξ)−g∗(0)] cos (ξρ)dξ= 1 +g∗(0) logρ, ρ∈(ρ0,ρ1), (7. 68) whereg∗(ξ) =1 2g(ξ),andf∗(ξ) =1 2f(ξ). Equations (7 .67) are easily reduced to those solved in the previous section by means of the transform ϕ= 2 arctan ( ρ) (see (7.36)), and we deduce σ1(y) =b/braceleftBigg K/parenleftBigg 2q1 2 1 +q/parenrightBigg/bracerightBigg−1 1 +q/radicalbig (y2−a2) (b2−y2), a<y<b, (7. 69) whereq=r2 0=a/b.It is interesting to compare this result with that for the cylindrically-shaped strips, given by (7 .46). Apart from a factor of1 4π, which ©200 1 CRC Press LLC is due to a diﬀerent deﬁnition of σ1,the result also coincides with that of Sneddon [55], once the complete elliptic integral identity [1] K/parenleftbigg2√q 1 +q/parenrightbigg = (1 +q)K(q) (7. 70) is taken into account. As might be expected, the expression for the capacitance per unit length for ﬂat strips has a similar format to that for cylindrically- shaped strips (7 .44) : C= 2K(κ) K/prime(κ), (7. 71) whereκ= (1−q)/(1 +q). The solution of the Equations (7 .68) may be approached in many ways. One approach is to reduce (7 .64) to triple integral equations with sine function ker- nels, and then to use the relationship between integral and series equations. A second way is to ﬁnd the relationship between integral and series equa-tions involving the cosine functions. Both approaches require rather bulkytransforms. However, a simpler way exploits the well-known mathematicaldevice employed in [55] .First, rescale the standard coordinates ( y /prime,z/prime), set- tingr=y/prime/b,z =z/prime/b, so that Equations (7 .68) become /integraldisplay∞ 0g(λ) cos (λr)dλ= 0, r∈(0,r0)∪(1,∞), (7. 72) /integraldisplay∞ 0λ−1[g(λ)−g(0)] cos (λr)dλ= 2 +g(0) logr, r∈(r0,1),(7. 73) wherer0=a/b. The mathematical device is a variant of the substitution method, and assumes that the unknown function ghas an expansion in a Neumann series g(λ) =∞/summationdisplay n=0anJ2n(λ), (7. 74) where {an}∞ n=0are the unknown coeﬃcients to be determined. The well- known discontinuous integral [19] ∞/integraldisplay 0J2n(ξ) cos (ξr)dξ=/parenleftbig 1−r2/parenrightbig−1 2T2n/parenleftBig/radicalbig 1−r2/parenrightBig H(1−r) (7. 75) shows that the integral Equation (7 .72) is satisﬁed automatically, for r∈ (1,∞). Substitution of (7 .74) into (7 .73),and use of another identity [19] (valid when n>0), ∞/integraldisplay 0ξ−1J2n(ξ) cos (ξr)dξ=1 2nT2n/parenleftBig/radicalbig 1−r2/parenrightBig , r< 1, ©200 1 CRC Press LLC leadstoth efollowingdualserie sequation sforth eunkn owncoeﬃcientsan:   ∞/summationtext n=0anT2n/parenleftbig√ 1−r2/parenrightbig =0,r ∈(0,r0), ∞/summationtext n=1n−1anT2n/parenleftbig√ 1−r2/parenrightbig =4+2a0logr,r ∈(r0,1).(7.76) Indeducing(7 .76),wehaveusedth eobviousrelationshi pa0=g(0). Theremainingstepsar enowobvious;th esubstitutioncos1 2ϕ=/radicalbig 1−r2 0 convertstheseequationstotrigonometricform ∞/summationdisplay n=1ancosnϕ=−A0,ϕ ∈(0,ϕ0), (7.77) ∞/summationdisplay n=1n−1ancosnϕ=4+2a0log/parenleftBig sinϕ 2/parenrightBig ,ϕ∈(ϕ0,π), wherecos1 2ϕ0=/radicalbig 1−r2 0.Followingth egenerals chem eoutlinedinSection 2.2weobtain ∞/summationdisplay n=1n−1anP(−1,0) n (cosϕ)=/braceleftbigg−2a0log/parenleftbig cos1 2ϕ/parenrightbig ,ϕ ∈(0,ϕ0) 4+2a0log/bracketleftbig1 2/parenleftbig 1+si nϕ 2/parenrightbig/bracketrightbig ,ϕ∈(ϕ0,π). (7.78) Astandar dcontinuityargumentestablishesthat a0=−2/braceleftbigg log/bracketleftbigg1 2/parenleftBig 1+si nϕ0 2/parenrightBig cosϕ0 2/bracketrightbigg/bracerightbigg−1 , (7.79) soth eﬁnalsolutionforth ecoeﬃcientsis am=−2a0P(−1,0) m (cosϕ0)+a0m/integraldisplayx0 −1P(−1,0) m (x)dx/parenleftBig 1+/radicalBig 1 2(1−x)/parenrightBig/radicalBig 1 2(1−x), (7.80) wherex0=cosϕ0.Thecapacitanc eofthetwostripsi sC=π 2g(0)=π 2a0,so that C=−π/braceleftbigg log/bracketleftbigg1 2/parenleftBig 1+si nϕ0 2/parenrightBig cosϕ0 2/bracketrightbigg/bracerightbigg−1 =−π/braceleftbigg log/bracketleftbigg1 2(1+r0)/radicalBig 1−r2 0/bracketrightbigg/bracerightbigg−1 ,(7.81) wherewerecallthat r0=a/b. Finallyletusconsiderth equadru polelenssyste moffou rchargedelectr odes, eachofwhi chisaﬂatstrip(se eFigure7.4).Usec oordinatesr,zsothatthe strips are separated by a distance 2 t= 2d/band comprise the four segments ©200 1 CRC Press LLC Figure 7.4 The charged thin-strip quadrupole. speciﬁed by r∈(−1,−r0)∪(r0,1),z=±1.The segments in the ﬁrst and third quadrants are positively charged to unit potential, whilst the remaining strips are negatively charged (to unit potential). The potential may be constructed as a sum of two dipole-like potentials in two ways. Group upper and lower pairs of strips, as dipole-like structures, so that ψ(r,z) =ψup(r,z) +ψlow(r,z) (7. 82) where ψup(r,z) =1 4π/integraldisplay1 r0log/bracketleftBigg (r+r/prime)2+/parenleftbig z−t 2/parenrightbig2 (r−r/prime)2+/parenleftbig z−t 2/parenrightbig2/bracketrightBigg σ(r/prime)dr/prime, (7. 83) ψlow(r,z) =−1 4π/integraldisplay1 r0log/bracketleftBigg (r+r/prime)2+/parenleftbig z+t 2/parenrightbig2 (r−r/prime)2+/parenleftbig z+t 2/parenrightbig2/bracketrightBigg σ(r/prime)dr/prime, (7. 84) σbeing the line charge density on that electrode in the ﬁrst quadrant ( r>0, z> 0). A variant grouping of the electrodes is vertical; however, both representations provide the same quadrupole potential distribution. We now construct the Fourier integral representation of the function ψ,in the three domains z >1 2t,|z|<1 2t, andz <−1 2t. First, use the Fourier transform representation [19] of the logarithmic function, 1 4log(a+b)2+p2 (a−b)2+p2=/integraldisplay∞ 0ξ−1e−ξpsin (ξa) sin (ξb)dξ, (7. 85) ©200 1 CRC Press LLCz rΨ Ψ ΨΨo oo o=1 =−1 =1 =−1 ot=d/b -1 r0 10-r valid when Re p>|Ima|+|Imb|.Also, as before, extend the domain of σto obtain a function σtot,deﬁned on (0 ,∞) by σtot(r/prime) =/braceleftbigg 0, r/prime∈(0,r0)∪(1,∞), σ(r/prime), r/prime∈(0,1),(7. 86) with Fourier sine transform representation σtot(r/prime) =/integraldisplay∞ 0f(λ) sinλr/primedλ. (7. 87) The desired representation is ψ(r,z) =/integraldisplay∞ 0λ−1f(λ) sinh/parenleftbigg λt 2/parenrightbigg e−λzsin (λr)dλ, z>t 2,(7. 88) ψ(r,z) =/integraldisplay∞ 0λ−1f(λ) sinh (λz)e−λt 2sin (λr)dλ,|z|<t 2,(7. 89) ψ(r,z) =−/integraldisplay∞ 0λ−1f(λ) sinh/parenleftbigg λt 2/parenrightbigg eλzsin (λr)dλ,z<−t 2.(7. 90) It is evident that the electrostatic potential deﬁned by (7 .88)–(7.90) is con- tinuous, including across the interfaces |z|=t/2,so the following triple inte- gral equations for the unknown function fhold: /integraldisplay∞ 0f(λ) sinλrdλ = 0, r∈(0,r0)∪(1,∞),(7. 91) /integraldisplay∞ 0λ−1/parenleftbig 1−e−λt/parenrightbig f(λ) sinλrdλ =−2, r∈(r0,1). (7. 92) The value of the discontinuous integral [19] ∞/integraldisplay 0J2n+1(λ) sinλrdλ =r/parenleftbig 1−r2/parenrightbig−1 2U2n/parenleftBig/radicalbig 1−r2/parenrightBig H(1−r) (7. 93) suggests the following Neumann series representation for f, f(λ) =∞/summationdisplay n=0bnJ2n+1(λ) ; (7. 94) it satisﬁes (7 .91) automatically when r>1.The remaining two integral equa- tions are transformed to the following dual series equations for the unknown coeﬃcients bn, ∞/summationdisplay n=0bnsin (2n+ 1)θ= 0, θ ∈(0,θ0),(7. 95) ∞/summationdisplay n=0(2n+ 1)−1bnsin (2n+ 1)θ=−2 +F(θ),θ∈/parenleftBig θ0,π 2/parenrightBig ,(7. 96) ©200 1 CRC Press LLC where cosθ=√ 1−r2, cosθ0=/radicalbig 1−r2 0, F(θ) =∞/summationdisplay n=0bnαn(θ) (7. 97) and αn(θ) =∞/integraldisplay 0λ−1e−λtJ2n+1(λ) sin (λsinθ)dλ. Using the expansion [1] sin (λsinθ) = 2∞/summationdisplay k=0J2k+1(λ) sin (2k+ 1)θ, (7. 98) we may write αn(θ) = 2∞/summationdisplay k=0βnksin (2k+ 1)θ, (7. 99) F(θ) = 2∞/summationdisplay n=0bn∞/summationdisplay k=0βnksin (2k+ 1)θ, where βnk=∞/integraldisplay 0λ−1e−λtJ2n+1(λ)J2k+1(λ)dλ. (7. 100) After the trivial substitution ϕ= 2θ(andϕ0= 2θ0) one obtains the follow- ing dual series equations on the standard domain (0 ,π), ∞/summationdisplay n=0bnsin/parenleftbigg n+1 2/parenrightbigg ϕ= 0, ϕ ∈(0,ϕ0), (7. 101) ∞/summationdisplay n=0/parenleftbigg n+1 2/parenrightbigg−1 bnsin/parenleftbigg n+1 2/parenrightbigg ϕ=−4 + 4∞/summationdisplay n=0bn∞/summationdisplay k=0βnksin/parenleftbigg k+1 2/parenrightbigg ϕ, ϕ∈(ϕ0,π). (7. 102) It should be noted that in the limiting case when t→0, the conjunction of oppositely charged strips eliminates sources to produce electrostatic ﬁeld. In this caseβnk=1 4/parenleftbig k+1 2/parenrightbig−1δnk. Equations (7 .101) and (7 .102) have a clear physical interpretation. The left- hand side of these equations represents ﬁeld terms for a single dipole pair of oppositely charged strips. The mutual coupling between the two dipoles isreﬂected in the presence of coupling terms on the right-hand side of (7 .102). The coeﬃcients β nkmeasure the strength of this coupling. The situation ©200 1 CRC Press LLC simpliﬁeswhentwopairsofstrip sarewellseparated,s othatt/greatermuch1.The coeﬃcie ntsβnkmaybeexpandedasarapidl yconvergentpowe rserie sint−1, equalling ∞/summationdisplay m=0(−1)mΓ(2n+2k+2m+3)Γ(2n+2k+2m+2)(2t)−2n−2k−2m−2 Γ(m+1)Γ(2k+m+2)Γ(2m+n+2)Γ(2n+2k+m+3). (7.103) Theexpansionisvalidfor t≤1 2,andconvergesrapidlyforlarge t. 7.4Axially-slottedellipticcylinders Inthissection ,weconsidercylindersofellipti ccross-sectionwithoneor twoapertures;thes estructure sareanalogue softh eslottedcircularcylin- dersconsidere dinprevioussections.Ellipticcylinde rcoordinate s(α,β,z ) weredeﬁne dinSection1.1.6(seeFigure7.5);brieﬂ y,intermsofCartesian coordinates,thecoordinate ssatisfy x=ccoshαcosβ,y =csinhαsinβ,z =z, (7.104) wheretherangeofparametersi s0≤α≤∞,and−π≤β≤π,thecoordinate surfacesα=α0=constant formafamilyofconf ocalellipti ccylinderswith semifocaldistance c=d 2,andthez-independe ntsolutions ψofLaplace’s equationsatisfy ∆ψ(α,β)=1 c2/parenleftbig cosh2α−cos2β/parenrightbig/bracketleftbigg∂2ψ ∂α2+∂2ψ ∂β2/bracketrightbigg =0. (7.105) Thegeometryofvariou sslottedellipticcylinderstobeconsideredar eshown inFigure7.6.Th eﬁrsttwo( Figure s7.6( a)and7.6(b ))areportionsofcoor- dinatesurfaces.Th elasttwoar ebette rdescri bedasportion sofcoordinate surfacesinavariantofellipticcylinderc oordinatestobedescribedlate rin thissection. Thegeneralsolutionofth eLaplac eequation ,givenbyasingle-laye rpo- tentialreprese ntation,i susedtoderiveth ebasi cseriesequationsforthese structures,showninFigur e7.6(a)–(d ). First,considertheelectrostaticﬁeldsurroundin gasingleellipti carccharged tounitpote ntial(Figure7.6(a)).Th epotentialhasasingle-l ayerrepresen- tation ψ(α,β) =−1 4π/integraldisplayβ0 −β0log/braceleftBig (x−x/prime)2+ (y−y/prime)2/bracerightBig σ(β/prime)dlβ, (7. 106) where it is to be understood that ( x,y) and (x/prime,y/prime) depend on ( α,β) and (α/prime,β/prime) according to (7 .104), dlβ=c/radicalbig cosh2α/prime−cos2β/primedβ/primeis the length dif- ©200 1 CRC Press LLC Figure 7.5 The elliptic cylinder coordinate system. ferential on the elliptic contour, and σ=σ(β/prime) is the line charge density. Simple algebra transforms the kernel in (7 .106) to the form log/braceleftBig (x−x/prime)2+ (y−y/prime)2/bracerightBig = 2 log (c/2) + log {2 [cosh (α+α/prime)−cos (β+β/prime)]} + log{2 [cosh (α−α/prime)−cos (β−β/prime)]}.(7. 107) The expansion of the logarithmic function in Fourier series [19] log (2 coshy−2 cosx) =y−2∞/summationdisplay n=1e−nycosnx n, y> 0 (7. 108) shows that the kernel (7 .107) is log/braceleftBig (x−x/prime)2+ (y−y/prime)2/bracerightBig = 2 log/parenleftbigg1 2cemax( α,α/prime)/parenrightbigg − 2∞/summationdisplay n=1n−1/braceleftBig e−n|α−α/prime|cosn(β−β/prime) +e−n(α+α/prime)cosn(β+β/prime)/bracerightBig .(7. 109) As before, introduce the function σtot,which extends the domain of σvia σtot(β/prime) =/braceleftbigg c/radicalbig cosh2α/prime−cos2β/primeσ(β/prime), β/prime∈(0,β0) 0, β/prime∈(β0,π)(7. 110) ©200 1 CRC Press LLC Figure 7.6 Various conﬁgurations of charged elliptic strips. with the understanding σtot(−β/prime) =σtot(β/prime); assume that σtotcan be repre- sented as a Fourier cosine series σtot(β/prime) =∞/summationdisplay m=0/parenleftbig 2−δ0 m/parenrightbig xmcosmβ/prime. (7. 111) Thus, (7.106) has the equivalent representation ψ(α,β) =−1 4π/integraldisplayπ 0K(α,β;α/prime,β/prime)σtot(β/prime)dβ/prime, (7. 112) where K(α,β;α/prime,β/prime) =−4∞/summationdisplay n=1n−1/braceleftBig e−n|α−α/prime|+e−n(α+α/prime)/bracerightBig cosnβcosnβ/prime + 4 log/parenleftbigg1 2cemax( α,α/prime)/parenrightbigg .(7. 113) ©200 1 CRC Press LLC By substitution of (7 .111) and (7 .113) into (7 .112), the form of the potential functionψin terms of unknown coeﬃcients xnis ψ(α,β) =∞/summationdisplay n=1n−1xn/bracketleftBig e−n|α−α/prime|+e−n(α+α/prime)/bracketrightBig cosnβ −x0log/parenleftbigg1 2cemax( α,α/prime)/parenrightbigg .(7. 114) The major and minor semi-axes are a=ccoshα/primeandb=csinhα/prime, so that1 2ceα/prime=1 2(a+b).When the elliptic arc degenerates to a circular arc (b→a), representation (7 .114) transforms to (7 .8).The only diﬀerence is the reference point, from which the potential is calculated. In order to make both representations compatible, redeﬁne ψ(α,β) as ψ(α,β) =−x0(α−α/prime) +∞/summationdisplay n=1n−1xn/bracketleftBig e−n|α−α/prime|+e−n(α+α/prime)/bracketrightBig cosnβ. (7. 115) Now use the obvious mixed boundary conditions to obtain the following dual series equations for the unknown Fourier coeﬃcients xn: ∞/summationdisplay n=1n−1xn/parenleftBig 1 +e−n2α/prime/parenrightBig cosnβ= 1, β ∈(0,β0), ∞/summationdisplay n=1xncosnβ=−1 2x0, β∈(β0,π).(7. 116) It is instructive to compare (7 .116) with its circular analogue (7 .11)–(7.12). Formally, the diﬀerence is the appearance of a new term ( e−n2α/prime), which is a measure of deviation between elliptic and circular strips. Equation (7 .116) is transformed in the usual way to the following second-kind Fredholm matrix equation, Xm−∞/summationdisplay n=1Xnκnm=γm, (7. 117) wherem= 1,2,...,Xm= (2/m)1 2xm, κnm=−e−2nα/prime{log [(1 −z0)/2]}−1ˆP(0,−1) n (z0)ˆP(0,−1) m (z0) nm +e−2nα/primeˆQ(−1,0) nm (z0) (7. 118) γm=−2{log [(1 −z0)/2]}−1ˆP(0,−1) m (z0) m, (7. 119) andz0= cosβ0.Furthermore, for these values of parameters, the normalised Jacobi polynomials ˆP(0,−1) n and ˆP(−1,0) n are deﬁned by ˆP(0,−1) n (x) = (2n)1 2P(0,−1) n (x),ˆP(−1,0) n (x) = (2n)1 2P(−1,0) n (x), ©200 1 CRC Press LLC andtheincompletescalarproductsare ˆQ(−1,0) nm (z0)=1/integraldisplay z0(1−z)−1ˆP(−1,0) n (z)ˆP(−1,0) m (z)dz. Thissecond-kindsystemmaybesolvednumericall yinth eusualway,employ- ingatruncationmethodthatisrapidlyconvergent. Theﬁeldoftheslotte dellipticcylinde rshowninFigure7.6(b),i nwhich theslotsar esymmetricall ylocate dandbothcharge dtouni tpositivepoten- tial,maybederi vedfromth esolutionobtaine daboveforthesingleelliptic strip.Takingintoconsiderationth echargeonbothstrips ,theFourie rseries representationforthepote ntialtake stheform ψ(α,β)=∞/summationdisplay n=1n−1x2n/bracketleftBig e−2n|α−α/prime|+e−2n(α+α/prime)/bracketrightBig cos2nβ −/braceleftbigg2x0(α−α/prime),(α>α 1) 0, (α<α 1)/bracerightbigg .(7.120) Satisfactionofth eboundarycondition sproduce sthedualserie sequations ∞/summationdisplay n=1n−1x2n/bracketleftBig 1+e−4nα/prime/bracketrightBig cosnϑ=1,ϑ∈(0,ϑ0)(7.121) ∞/summationdisplay n=1x2ncosnϑ=−1 2x0,ϑ ∈(ϑ0,π)(7.122) whereϑ=2βandϑ0=2β0. Thesolutionisreadil yderivedfro m(7.117)withth efollowingchanges :in thematrixeleme ntsκnm,thefactore−2nα/primeisreplace dbye−4nα/prime,theparame- terz0isreplacedby2 z2 0−1=cos2β0,andtheunknown xmisreplace dbyx2m. Thiscompletesthesolutionforth eslotte dellipticcylinderwithide ntically chargedcomponents. Theﬁeldoftheslottedellipticcylinde rshowninFigur e7.6(b )inwhich the slots are symmetrically located, but are oppositely charged (each to unit potential), may be derived from the representation ψ(α,β) =∞/summationdisplay n=0x2n+1/parenleftbig n+1 2/parenrightbig/bracketleftBig e−(2n+1)|α−α/prime|+e−(2n+1)(α+α/prime)/bracketrightBig cos(2n+ 1)β, (7. 123) ©200 1 CRC Press LLC wherethecoeﬃcie ntsx2n+1satisf ythedualseriesequations ∞/summationdisplay n=0x2n+1/parenleftbig n+1 2/parenrightbig/bracketleftBig 1+e−(4n+2)α/prime/bracketrightBig cos/parenleftbigg n+1 2/parenrightbigg ϑ=1,ϑ∈(0,ϑ0), (7.124) ∞/summationdisplay n=0x2n+1cos/parenleftbigg n+1 2/parenrightbigg ϑ=0,ϑ∈(ϑ0,π), (7.125) andthevariabl eϑandparamete rϑ0arethesameasinEquation s(7.121). Itcanbereadil yshownthatbyreplacin gϑbyπ−θandidentifyin gbnwith (−1)nx2n+1,Equation s(7.124)reduc etoequation softhesametypeas(7.94) and(7.95);however,thetermontheright-han dsideof(7.124)hasamuch simple ranalytica lstructure. Suppressin gtheintermediat esteps ,theﬁnalformofthesyste mis x2m+1+∞/summationdisplay n=0x2n+1e−(4n+2)α/primeRnm(z0)=Cm, (7.126) wherem=0,1,2,..., Cm=Pm(z0)/Q−1 2(z0), (7.127) Rnm(z0)=/parenleftbig m+1 2/parenrightbig2 n+1 2/braceleftBigg Pn(z0) Q−1 2(z0)/integraldisplayz0 −1Q−1 2(z)Pm(z)dz+Q(0,0) nm(z0)/bracerightBigg , (7.128) andz0=cosϑ0=cos2β0.Theintegral sappearin gin(7.128)maybeeasily evaluate d(seeAppendix ,(B.97)). Theconﬁguration softhecharge dellipti cstripsshowninFigure s7.6(c)and 7.6(d)arebestdescri bedbyoblateellipti ccylinde rcoordinate s(seeFigure 7.7);thissyste misobtaine dbyreplacin gtheparamete rβbyπ 2−βin the prolate variant of elliptic cylinder coordinates deﬁned at the beginning of this section. Thus, x= coshαsinβ, y =csinhαsinβ, z =z, where the range of parameters is 0 ≤α<∞,−π≤β≤π,and the Laplacian forz-independent potentials is /triangleψ(α,β) =1 c2/parenleftbig cosh2α−sin2β/parenrightbig/bracketleftbigg∂2ψ ∂α2+∂2ψ ∂β2/bracketrightbigg . (7. 129) By simple algebra, one may verify that representation of the potential is given by (cf. (7.115)) ψ(α,β) =−x0(α−α/prime) +∞/summationdisplay n=1n−1xn/bracketleftBig e−n|α−α/prime|+ (−1)ne−n(α+α/prime)/bracketrightBig cos 2nβ (7. 130) ©200 1 CRC Press LLC Figur e7.7 Theoblat eellipti ccylinde rcoordinat esystem. wheretheexpansio n(7.111)remain strueforthemodiﬁeddeﬁnitio nforσtot, σtot(β/prime)=/braceleftbigg c/radicalbig cosh2α/prime−sin2β/primeσ(β/prime),β/prime∈(0,β0), 0, β/prime∈(β0,π).(7.131) Thepotentialfunctio nfortheellipti cstripsshowninFigure s7.6(c)and 7.6(d)maybereadil yderivedfromthesolution salread yobtaine dinthis sectio nwithafewsimpl emodiﬁcations .Forthesingleellipti cstrip(Fig- ure7.6(c)),multipl ythematri xeleme ntsκnm(7.118)byafacto rof(−1)n. Forthepairofsymmetricall ylocatedstrips(Figur e7.6(d))bothpositively charge dtounitpotential,nochange sareneeded ;however,ifthepairofsym- metricall ylocatedstrips(Figur e7.6(d))areoppositel ycharged ,changethe sign of the term containing e−(4n+2)α/prime, replacing it by −e−(4n+2)α/prime. The line charge density σcan now be calculated using deﬁnition (7 .131). 7.5 Slotted cylinders of arbitrary proﬁle The study of the slotted elliptical cylinder suggests that the idea of regu- larisation might beneﬁcially be extended to determine the potential of more general two-dimensional, thin, charged conductors. In examining the ellipticcylinder, we analytically inverted that part of the series equations (see (7.116)) that deﬁnitely corresponds to a circular proﬁle. From the perspective ofthe method of regularisation a singular part of the operator associated with ©200 1 CRC Press LLC the series equations formulation was inverted. The remaining contributions (visible as the terms proportional to e−n2α/prime) are regular (analytic) perturba- tion terms that measure the deviation of the elliptic proﬁle from the circular. The purpose of this section is to show how a regularisation approach may be extended to open (slotted) hollow cylinders with arbitrarily proﬁled cross-section. Although we do not aim to compute the electrostatic ﬁelds of allpossible conﬁgurations, nevertheless we wish to demonstrate how the meth-ods developed for canonical conductors work in the wider context. In par-ticular, the regularised system of equations for an open cylinder of arbitrarycross-section with one slot or aperture will be obtained. This approach has been developed by Tuchkin [52, 65] in the context of a rigorous treatment of diﬀraction by open thin cylinders of arbitrary cross-section. The starting point is the construction of the solution to the Dirichlet bound- ary problem for Laplace equation on an arc of a hollow circular cylinder ofunit radius. In cylindrical polar coordinates ( r,ϕ) the electrostatic potential ψproduced by such a thin strip with (as yet unknown) charge density σis given by the single-layer potential of the type (7. 2), ψ(r,ϕ) =−1 4π/integraldisplayϕ0 −ϕ0log/vextendsingle/vextendsingle1−2rcos(ϕ−ϕ/prime) +r2/vextendsingle/vextendsingleσ(ϕ/prime)dϕ/prime. (7. 132) If the conductor is charged to potential ψ0(ϕ) (as a function of position), enforcement of the boundary condition ψ(1,ϕ) =ψ0(ϕ),ϕ∈[−ϕ0,ϕ0], (7. 133) produces the ﬁrst-kind Fredholm equation −1 2π/integraldisplayϕ0 −ϕ0log/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 sinϕ−ϕ/prime 2/vextendsingle/vextendsingle/vextendsingle/vextendsingleσ(ϕ/prime)dϕ/prime=ψ0(ϕ), ϕ∈[−ϕ0,ϕ0].(7. 134) Extend the domain of deﬁnition of the line charge density σto a function σ∗(ϕ/prime) =/braceleftbigg σ(ϕ/prime), ϕ/prime∈[−ϕ0,ϕ0] 0, ϕ/prime∈[−π,π]\[−ϕ0,ϕ0], (7. 135) so that the function σ∗satisﬁes −1 2π/integraldisplayπ −πlog/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 sinϕ−ϕ/prime 2/vextendsingle/vextendsingle/vextendsingle/vextendsingleσ∗(ϕ/prime)dϕ/prime=ψ0(ϕ), ϕ∈[−ϕ0,ϕ0].(7. 136) The logarithmic kernel has a Fourier series expansion log/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 sinϕ−ϕ /prime 2/vextendsingle/vextendsingle/vextendsingle/vextendsingle=−1 2n=−∞/summationdisplay n=−∞/prime1 |n|ein(ϕ−ϕ/prime), (7. 137) ©200 1 CRC Press LLC whereprimeindicatesomissionofthezer oindex(n=0)term.Assumethat bothσ∗andψ0haveFourierseriesexpansions σ∗(ϕ/prime)=∞/summationdisplay n=−∞xneinϕ/prime, (7.138) 2ψ0(ϕ)=∞/summationdisplay n=−∞fneinϕ, (7.139) where {xn}∞ n=−∞areunknowncoeﬃcientstobedetermined,bu tthecoeﬃ- cients {fn}∞n=−∞arekn own.Substitutionofthes eexpansion sproduce sthe followingdualseriesequations, ∞/summationdisplay n=−∞/prime1 |n|xneinϕ=∞/summationdisplay n=−∞fneinϕ,ϕ∈[−ϕ0,ϕ0], (7.140) ∞/summationdisplay n=−∞xneinϕ=0,ϕ∈[−π,π]\[−ϕ0,ϕ0], (7.141) wheretheprim einEquation(7.140)mean sthattheindexzerotermis omittedfromth esummation. AsshowninSectio n2.2,th ecanonical equations(7.140)and(7.141)are solvableanalytically,andthusprovideastartingpoi ntforthegeneralisation toslotte dcylinder sofarbitrarycross-section. LetLdenotethetwo-dimensionalcross-sectioninthe xyplaneofthear- bitrarilyshaped,inﬁnitelythin,slotte dconductor(seeFigur e7.8).Itwill be assumed to be suﬃciently smooth; the precise degree of smoothness will become apparent below. Let p=p(x,y) denote a point on the contour L. If the conductor is charged to the potential ψ0(p) (at eachp∈L), the Dirichlet boundary conditions to be enforced at each point p∈Lare ψ(p−0) =ψ(p+ 0) =ψ0(p). (7. 142) The potential ψ(q) of the electrostatic ﬁeld produced by this conductor has a single-layer potential representation (7. 1) in terms of the surface charge densityσ, ψ(q) =−1 2π/integraldisplay Llog(|p−q|)σ(p)dlp, q∈R2, (7. 143) wheredlpis the diﬀerential of arc length at the point p∈L,qis a point at which the electrostatic potential is considered, and R=|p−q|is the distance between the point pon the conductor and the observation point q. Applying the boundary condition (7. 142) to Equation (7. 143) yields the integral equation 1 2π/integraldisplay Llog(|p−q|)σ(p)dlp=−ψ0(q), q∈L (7. 144) ©200 1 CRC Press LLC Figure 7.8 Cross-section of the arbitrarily shaped, inﬁnitely thin, slotted con- ductor. for the unknown surface charge density σ. Once this is found, the electrostatic potential at any point qcan be found from (7. 143), and all the relevant physical quantities such as charge and capacitance are easily calculated. Our reformulation of the integral Equation (7. 144) begins by regarding the open contour Las part of a larger closed structureS, which is parametrised by the functions x(θ),y(θ) whereθ∈[−π,π]; the parametrising functions are pe- riodic so that x(−π) =x(π),y(−π) =y(π). The contour Lis parametrised by the subinterval [ −θ0,θ0], L={(x(θ),y(θ)),θ∈[−θ0,θ0]}, and the aperture is created by the removal from Sof the segment L/prime={(x(θ),y(θ)),θ∈[−π,−θ0]∪[θ0,π]}. In order to employ the regularisation procedure to be described, the param- etrisation of the contour Smust be continuous and twice diﬀerentiable at each point pofS. Moreover, the computational eﬀectiveness of the numeri- cal algorithm derived from the regularised system increases as the degree ofcontour smoothness (diﬀerentiability) increases. With this parametrisation, the diﬀerential of arc length is l(τ) =/radicalbig (x/prime(τ))2+ (y/prime(τ))2, ©200 1 CRC Press LLCoy xp(x( ),y( )) q(x( ),y( ))θp θp θq θq SL andthei ntegralEquation(7.144)take stheform 1 2π/integraldisplayθ0 −θ0log(R)σ0(τ)dτ=−ψ0(θ),θ∈[−θ0,θ0], (7.145) whereR(θ,τ)=/radicalBig [x(θ)−x(τ)]2+[y(θ)−y(τ)]2isthedistancebetweenpoints oftheco ntourparametrisedby θandτ,σ0(τ)=σ(x(τ),y(τ))l(τ),and ψ0(θ)=−ψ0(x(θ),y(θ)). Introducethenewunknownfunction z,extendingth edomai nofσ0and deﬁnedby z(τ)=/braceleftbiggσ0(τ),τ ∈[−θ0,θ0], 0,τ ∈[−π,−θ0]∪[θ0,π].(7.146) Transform(7.145)toanintegralequationforthisnewunknownove rthefull interval[ −π,π]ofth eangularcoordinat eθ: 1 2π/integraldisplayπ −πlog(R)z(τ)dτ=−Ψ0(θ),θ∈[−θ0,θ0]. (7.147) Equation(7.147),togetherwiththerequirementthat zvanishesoutsid ethe interval[ −θ0,θ0],iscompletelyequivale nttoEquatio n(7.145). WenowconvertEquation(7.147)toadualserieswithatrigonometric kernel.Thefunction zisreprese ntedbyitsFourierseries,whils tthekernel of(7.147)isexpandedasadoubleFourierseries.Thesemi-inversionand regularisationofdualserieswithtrigonometricfunction skernelsdescri bedin Chapter2istheke ytechnicalideauponwhichthismeth odrelies. The ﬁrst stage is to obtain the integral equation in the equivalent form of a dual series equation with exponential functions einθ.Split the kernel of the integral Equation (7. 147) into singular and regular parts: log(R(θ,τ)) = ln(2/vextendsingle/vextendsingle/vextendsingle/vextendsinglesinθ−τ 2/vextendsingle/vextendsingle/vextendsingle/vextendsingle) +H(θ,τ). (7. 148) The singular part of the kernel (7. 148) has the expansion (7. 137). Our as- sumptions about the surface Simply thatH(θ,τ) is smooth and continuously diﬀerentiable with respect to θandτ; this allows its expansion in a double Fourier series, H(θ,τ) =∞/summationdisplay p=−∞∞/summationdisplay n=−∞hnpei(nθ+pτ), θ,τ ∈[−π,π], (7. 149) where∞/summationdisplay p=−∞∞/summationdisplay n=−∞(1 +|p|2)(1 +|n|2)|hnp|2<∞. ©200 1 CRC Press LLC Hereth ecoeﬃcients hnparegive nby hnp=1 4π2/integraldisplayπ −π/integraldisplayπ −πH(θ,τ)e−i(nθ+pτ)dθdτ. (7.150) Representtheconductorpote ntialfunction ψ0andth eunknown zintheir Fourierseries: 2ψ0(θ)=∞/summationdisplay n=−∞gneinθ,θ ∈[−π,π](7.151) z(τ)=∞/summationdisplay n=−∞ςneinτ,τ ∈[−π,π]. (7.152) Inserting(7.137)an d(7.148)–(7.152)int oEquation(7.147)an drecalling thatz(τ)vanishesoutsideth einterval[ −θ0,θ0],weobtainth efollowin gdual seriesequation swithexponentialkernels: ∞/summationdisplay n=−∞/prime|n|−1ςneinθ−2∞/summationdisplay n=−∞einθ∞/summationdisplay p=−∞hn,−pςp=∞/summationdisplay n=−∞gneinθ, θ∈[−θ0,θ0],(7.153) ∞/summationdisplay n=−∞ςneinθ=0,θ ∈[−π,−θ0]∪[θ0,π]. (7.154) ThustheintegralEquation(7.147)isconverte dtoequivalentdualseries equationsdeﬁnedontwosubinter valsof[ −π,π],withunknowns {ςn}∞ n=−∞to befound. Followingthepr ocedureofSection2.2,weconvertthissyste monewit hreal trigonometric kernels. Introduce the new unknowns xn= (ζn+ζ−n)/|n|, y n= (ζn−ζ−n)/|n|, (7. 155) wheren= 1,2,.... Set g+ n=gn+g−n, g− n=gn−g−n,(n= 1,2,...), (7. 156) and deﬁne the matrices from the coeﬃcients {hnp}∞ n,p=−∞(7. 150) by k(++) np = [(hn,p+hn,−p) + (h−n,p+h−n,−p)]/(2 + 2δn0), n,p ≥0; k(+−) np = [(hn,p−hn,−p) + (h−n,p−h−n,−p)]/(2 + 2δn0), n≥0,p≥1; k(−+) np = [(hn,p+hn,−p)−(h−n,p+h−n,−p)]/2, n ≥1,p≥0; k(−−) np = [(hn,p−hn,−p)−(h−n,p−h−n,−p)]/2, n,p ≥1. (7. 157) ©200 1 CRC Press LLC Wemaytherefor ereduceth esyste mofEquations(7.154)totwocoupled systemsofdualserie sequation swithtrigonometricfunctionkernels, ∞/summationdisplay n=1xncosnθ=a0+∞/summationdisplay n=1ancosnθ,θ ∈[0,θ0], ∞/summationdisplay n=1nxncosnθ=−ζ0,θ ∈[θ0,π], (7.158) and ∞/summationdisplay n=1ynsinnθ=∞/summationdisplay n=1cnsinnθ,θ ∈[0,θ0], ∞/summationdisplay n=1nynsinnθ=0,θ ∈[θ0,π], (7.159) where a0=g0+2k(++) 00ζ0+2∞/summationdisplay p=1p(k(++) 0pxp−k(+−) 0pyp), an=g+ n+2k(++) n0ζ0+2∞/summationdisplay p=1p(k(++) npxp−k(+−) npyp), cn=g− n+2k(−+) n0ζ0−2∞/summationdisplay p=1p(k(−−) npyp−k(−+) npxp). (7.160) Theseequationsaren owinstandardformtoapplyth eresult sofSection 2.2,andwewritedowntheregularise dsystemoflinearequationsobtaine dby this process. It produces two coupled matrix equations (of second kind) with the rescaled unknowns Xn=xn√ 2n, Y n=yn√ 2n, X 0= 2ζ0. (7. 161) Settingt0= cosθ0, the systems are Ym+∞/summationdisplay p=1/radicalbig 2p∞/summationdisplay n=1√ 2n/bracketleftBig Ypk(−−) np−Xpk(−+) np/bracketrightBig ˆQ(0,1) n−1,m−1(t0) =∞/summationdisplay n=1√ 2n(X0k(−+) np +g− n)ˆQ(0,1) n−1,m−1(t0),(7. 162) and Xm−∞/summationdisplay p=1/radicalbig 2p∞/summationdisplay n=1√ 2n/bracketleftBig Xpk(++) np−Ypk(+−) np/bracketrightBig ˆQ(1,0) n−1,m−1(t0) =∞/summationdisplay n=1√ 2n(X0k(++) n0+g+ n)ˆQ(1,0) n−1,m−1(t0) +X0(1 +t0)1 mˆP(0,1) m−1(t0),(7. 163) ©200 1 CRC Press LLC wherem= 1,2,...; an additional equation, which is to be solved together with the Equations (7. 162) and (7. 163), is ∞/summationdisplay p=1/radicalbig 2p(Xpk(++) 0p−Ypk(+−) 0p)+ (1 +t0) 2∞/summationdisplay p=1/radicalbig 2p∞/summationdisplay n=1/radicalbigg 2 n/bracketleftBig Xpk(++) np−Ypk(+−) np/bracketrightBig ˆP(0,1) n−1(t0) =−(1 +t0) 2∞/summationdisplay n=1/radicalbigg 2 n(X0k(++) n0+g+ n)ˆP(0,1) n−1(t0) −g0−X0/bracketleftbigg k(++) 00 +1 2ln((1−t0) 2)/bracketrightbigg .(7. 164) Here ˆQ(0,1) n,m(t0) =/integraldisplay1 t0(1 +t)ˆP(0,1) n(t)ˆP(0,1) m(t)dt is the usual normalised incomplete scalar product. This regularised system of equations is a coupled Fredholm matrix system of second kind, which may be satisfactorily solved by the usual process of trun- cation. In addition to the standard considerations about truncation number,some attention must be paid to the rate of convergence of the double Fourier series representation (7. 149) of the distance between points on the cylinder proﬁle. With this proviso, the regularisation approach and the resulting sys-tem of equations provides a satisfactory basis for numerical computations ofthe electrostatic ﬁelds surrounding open (singly-slotted) hollow cylinders witharbitrarily proﬁled cross-sections. ©200 1 CRC Press LLC Chapter8 MoreComplicatedStructures Inthischapter,weconsideraclassofstructure swhi ch,fromatechnical pointofview,ismor ecomplicatedthanthos eclassesexaminedinprevious chapters.Theclasscomprisesplates ,som eofsimplegeometrico rcanonical shape ,andotherswithagreate rdegre eofcomplexity.Complexityisarelati ve notion.Th edeterminationofthepote ntialforanelectriﬁe dcirculardiscisnot complicated,an ditssolutio nhasbeenknownforalon gtim e[8];however,the analogousproblemforanelectriﬁe dellipticplateseemstobemorecomplex, andit srigorou ssolutionhasbee nobtainedonlycomparativelyrece ntly[5]. Inthesameway,thepotentialassociatedwithacharge dthinsphericalshell, withanellipti chole ,orwithth echargedsphericallyconformalellipticplate, provide sproblemsofequalcomplexity.Rathe rmorecomple xareproblems generatedbycrossedplates,orbypolygonplates,etc.I nthishierar chy, arbitrarily-shapedﬂatplate sprese ntth emos tcomplexproblemstructures foranalyticalmethods. Inthischapterweoutlinehowth eintegralmethodsmaybeusedfora uniﬁedtreatmentofdeterminingthepote ntialforallthes echargedstructures, fromth eelectriﬁeddisctoarbitrarily-shape dcharge dﬂatplates.Thecircular andellipticdiscsar econsideredinSection s8.1and8.2,respectively;this formsthebasisforcalculatin gthecapacitanc eofaspherically-curvedelliptic plate.Platesthatar eregularpolygon sareexaminedinSection8.3.The ﬁniterectangularstripisconsideredinSection8.4;considerablemanipulation isrequire dtodemonstratethatth eregularise dsyste misindee ddominantly diagonal.Inth eﬁnalsection(8.5)wecalculateth ecapacitanceofa coupl ed pairofchargedconductors ,thesphericalcapandth ecirculardisc .This exampleisi nterestin gbecaus ethecomponentsarepartsofcoordinatesurfaces belongin gto diﬀerent coordinatesystems,andth eresultantequation sare particularcasesofthe integro-seriesequation sbrieﬂydescribedinSection 2.9. ©200 1 CRC Press LLC Figur e8.1 TheﬂatplateS0witharbitrarily-sha pedboundar yΓ.Itliesonthe xOyplaneandhascompleme ntS1. 8.1Rigorou ssolutio nmeth odsforcharge dﬂatplates Inthissectio nweexamin ethecanonica lproble mofanelectriﬁe dcircular disc;itprovidesastartin gpointforthegeneralisatio nofintegra lmeth odsto moregenera lstructures. Conside ranarbitrarily-sha pedﬂatplateoccupyingtheﬁnitesurfac eregion S0intheplanez=0(seeFigur e8.1);letS1bethe(unbounded )comple- mentary part of this plane and Γ be its boundary contour. The potential generated by the structure may be represented in the form of the single-layer potential, ψ(x,y,z ) =−1 4π/integraldisplay/integraldisplay S0σ(x/prime,y/prime)dx/primedy/prime /radicalBig (x−x/prime)2+ (y−y/prime)2+z2, (x,y,z )∈R3\Γ (8. 1) whereσis surface charge density induced when the structure is immersed in a known potential ﬁeld ψ0; enforcement of the boundary condition ψ=ψ0(− →r),− →r∈S0 provides an integral equation determining σ. Recall that the inverse distance /vextendsingle/vextendsingle/vextendsingle− →r−− → r/prime/vextendsingle/vextendsingle/vextendsingle−1 =/braceleftBig (x−x/prime)2+ (y−y/prime)2+ (z−z/prime)2/bracerightBig−1 2(8. 2) ©200 1 CRC Press LLCz y xS0Ψο(x,y)S1Γ is a fundamental solution of the Laplace equation in R3. In order to represent ψas a double Fourier transform, we employ the representation (1. 206) of the inverse distance that is discontinuous in z, /braceleftBig (x−x/prime)2+ (y−y/prime)2+z2/bracerightBig−1 2 =1 2π/integraldisplay∞ −∞dνcosν(x−x/prime)/integraldisplay∞ −∞dµcosµ(y−y/prime)/radicalbig ν2+µ2e−√ ν2+µ2|z|.(8. 3) Then extend to domain of deﬁnition of σto the whole of the plane z= 0 via σt(x/prime,y/prime) =/braceleftbiggσ(x/prime,y/prime),(x/prime,y/prime)∈S0, 0, (x/prime,y/prime)∈S1.(8.4) For the most general structures, the double Fourier transform of the function σtmay be expressed in terms of four unknown functions f, g, h, andt, σt(x,y) =/integraldisplay∞ −∞dνcosνx/integraldisplay∞ −∞dµ{f(ν,µ) cosµy+h(ν,µ) sinµy} +/integraldisplay∞ −∞dνsinνx/integraldisplay∞ −∞dµ{g(ν,µ) cosµy+t(ν,µ) sinµy}.(8. 5) The boundary condition ψ(x,y,+0) =ψ(x,y,−0) =ψ0(x,y), (x,y)∈S0 (8. 6) now provides an integral equation for the extended surface charge density σt, −1 4π/integraldisplay∞ −∞/integraldisplay∞ −∞σt(x/prime,y/prime)/radicalBig (x−x/prime)2+ (y−y/prime)2dx/primedy/prime=ψ0(x,y),(x,y)∈S0. (8. 7) Substitution of the double Fourier transforms (8 .5) and (8. 3) for the func- tionsσtand inverse distance, respectively, together with the recognition that σtvanishes on S1,produces the dual integral equations −π 4/integraldisplay∞ −∞dνcosνx/integraldisplay∞ −∞dµ (ν2+µ2)1 2{f(ν,µ) cosµy+h(ν,µ) sinµy} −π 4/integraldisplay∞ −∞dνsinνx/integraldisplay∞ −∞dµ (ν2+µ2)1 2{g(ν,µ) cosµy+t(ν,µ) sinµy} =ψ0(x,y), (x,y)∈S0,(8. 8) and /integraldisplay∞ −∞dνcosνx/integraldisplay∞ −∞dµ{f(ν,µ) cosµy+h(ν,µ) sinµy} +/integraldisplay∞ −∞dνsinνx/integraldisplay∞ −∞dµ{g(ν,µ) cosµy+t(ν,µ) sinµy} = 0, (x,y)∈S1.(8. 9) ©200 1 CRC Press LLC Equations (8 .8) and (8.9) describe the most general electrostatic ﬁeld for an arbitrarily-shaped charged ﬂat plate. To consider the special case of a charged circular disc, introduce the parametrisation by x=ρcosφ, y =ρsinφ,so that the bounding contour Γ is ρ= 1.Then, setting τ=/radicalbig ν2+µ2, use the expansion sinseries(derivedfromthegeneratin gseries ,seeAppendix ,(B. 139)–(B. 142)), cos (νx) cos (µy) = cos (νρcosφ) cos (µρsinφ) =∞/summationdisplay m=0/parenleftbig 2−δ0 m/parenrightbig J2m(τρ)T2m/parenleftbig µτ−1/parenrightbig T2m(cosφ),(8. 10) cos (νx) sin (µy) = cos (νρcosφ) sin (µρsinφ) = 2ντ−1sinφ∞/summationdisplay m=0J2m+1(τρ)U2m/parenleftbig µτ−1/parenrightbig U2m(cosφ),(8. 11) sin (νx) cos (µy) = sin (νρcosφ) cos (µρsinφ) = 2ντ−1∞/summationdisplay m=0J2m+1(τρ)U2m/parenleftbig µτ−1/parenrightbig T2m+1(cosφ),(8. 12) sin (νx) sin (µy) = sin (νρcosφ) sin (µρsinφ) = 2ντ−1sinφ∞/summationdisplay m=0J2m+2(τρ)U2m+1/parenleftbig µτ−1/parenrightbig U2m+1(cosφ).(8. 13) If the given potential ψ0(x,y) is representable as a trigonometric series (or equivalently as a series in the Chebyshev polynomials Tm(cosφ) and Um(cosφ)), then using the orthogonality of the even or odd Chebyshev poly- nomials on/parenleftbig 0,π 2/parenrightbig as appropriate, one may deduce dual integral equations, involving the Bessel function kernels of the form Jm/parenleftBig/radicalbig ν2+µ2ρ/parenrightBig for the unknownsf, g, h, andt. In the simplest case, suppose that the circular disc is raised to unit potential so thatψ◦(x,y) = 1 onS0.The bivariate dual integral equations become /integraldisplay∞ 0dν/integraldisplay∞ 0dµF(ν,µ)J0/parenleftBig/radicalbig ν2+µ2ρ/parenrightBig = 1,0≤ρ<1 (8. 14)/integraldisplay∞ 0dν/integraldisplay∞ 0dµ/radicalbig ν2+µ2F(ν,µ)J0/parenleftBig/radicalbig ν2+µ2ρ/parenrightBig = 0, ρ> 1 (8. 15) where the as yet unknown function Frepresents the electrostatic potential by ψ(x,y) =/integraldisplay∞ 0dνcosνx/integraldisplay∞ 0dµF(ν,µ)e−√ ν2+µ2|z|cos (µy). (8. 16) ©200 1 CRC Press LLC Foracirculardisc,itisobviou sthatFdependsonlyupon τ=/radicalbig ν2+µ2, sothatF(ν,µ)=F/parenleftBig/radicalbig ν2+µ2/parenrightBig =F(τ),andth eduali ntegralequations become /integraldisplay∞ 0dτ.τF (τ)J0(τρ)=2 π,0≤ρ<1, (8.17) /integraldisplay∞ 0dτ.τ2F(τ)J0(τρ)=0,ρ> 1. (8.18) Thesolutioni sgivenby(se eSection2.6) F(τ)=4π−2τ−2sinτ,so F(ν,µ) =4 π2/parenleftbig ν2+µ2/parenrightbig−1sin/parenleftBig/radicalbig ν2+µ2/parenrightBig . (8. 19) Thesubstitution method provides an alternative and very useful method for solving Equations (8 .14) and (8.15). Seek the solution Fas an expansion in the Neumann series F(ν,µ) =/parenleftbig ν2+µ2/parenrightbig−3 4∞/summationdisplay k=0xkJ2k+1 2/parenleftBig/radicalbig ν2+µ2/parenrightBig (8. 20) where the coeﬃcients xkare to be found. Insertion of (8 .20) into (8.14) and (8.15) yields (using again the substitution τ=/radicalbig ν2+µ2) ∞/summationdisplay k=0xk/integraldisplay∞ 0τ−1 2J0(τρ)J2k+1 2(τ)dτ=2 π,0≤ρ<1, (8. 21) ∞/summationdisplay k=0xk/integraldisplay∞ 0τ1 2J0(τρ)J2k+1 2(τ)dτ= 0, ρ> 1. (8. 22) The integrals occurring in (8 .21) and (8.22) have the values [14] /integraldisplay∞ 0τ−1 2J0(τρ)J2k+1 2(τ)dτ= 2−1 2Γ/parenleftbig k+1 2/parenrightbig Γ (k+ 1)P2k/parenleftBig/radicalbig 1−ρ2/parenrightBig , (8. 23) when 0 ≤ρ<1,and /integraldisplay∞ 0τ1 2J0(τρ)J2k+1 2(τ)dτ= 21 2Γ (k+ 1) Γ/parenleftbig k+1 2/parenrightbigP2k/parenleftBig/radicalbig 1−ρ2/parenrightBig /radicalbig 1−ρ2H(1−ρ). (8. 24) Whenρ > 1,the integrals occurring in (8 .22) therefore vanish identically for eachk, so that the equation is satisﬁed automatically; when 0 ≤ρ <1, Equation (8 .21) becomes ∞/summationdisplay k=0Γ/parenleftbig k+1 2/parenrightbig Γ (k+ 1)xkP2k/parenleftBig/radicalbig 1−ρ2/parenrightBig =2√ 2 π. (8. 25) ©200 1 CRC Press LLC Figur e8.2 Thecharge dellipti cdisc. Becaus etheevenorderLegendr epolynomial sareorthogona lon(0,1), /integraldisplay1 0ρ/parenleftbig 1−ρ2/parenrightbig−1 2P2k/parenleftBig/radicalbig 1−ρ2/parenrightBig P2n/parenleftBig/radicalbig 1−ρ2/parenrightBig dρ=(4n+1)−1δkn, (8.26) wemaydeduce xn=(2/π)3 2δn0(n=1,2,...). (8.27) Thus F(τ)=τ−3 2(2/π)3 2J1 2(τ)=4π−2τ−2sinτ, inagreeme ntwiththepreviousl yobtaine dresult(8.19). 8.2Thecharge dellipti cplate Aswellasitsownintrinsi cinterest ,thecalculatio nofelectrostati cpotential duetoacharge dellipti cplatedemonstrate sbasicstepsofamoregeneral meth odtocalculat ethepotentialofaﬂatchargeplateofarbitrar yshape. Thefundame ntalideaistouseaparametrisatio nthatreduce stheoriginal proble mtodisc-lik eequation swithdisc-lik esolutions. Guide dbytheresult softhepreviou ssection ,letusconside rtheproblem inCartesia ncoordinate s(seeFigur e8.2).Whentheplateischarge dtounit potential (ψ0= 1 onS0), the form of the potential to be found is also given ©200 1 CRC Press LLCa -a oz y xb-bΨ o(x,y)=1 by (8.16). It should be noted that this simpler form is the result of symmetry. Ifaandbdenote the minor and major semi-axes, respectively, introduce the coordinates x=bρcosφ, y =aρsinφ (8. 28) so that the boundary of the elliptic plate Γ is given by ρ= 1.Letq=a/b,so thatq≤1. Use the boundary conditions (8. 6) to obtain the dual integral equations for the unknown function F(ν,µ),valid forφ∈/parenleftbig 0,1 2π/parenrightbig , /integraldisplay∞ 0dνcos (νbρcosφ)/integraldisplay∞ 0dµF(ν,µ) cos (µaρsinφ) = 1,0≤ρ<1, (8. 29)/integraldisplay∞ 0dνcos (νbρcosφ)/integraldisplay∞ 0dµ/radicalbig ν2+µ2F(ν,µ) cos (µaρsinφ) = 0, ρ> 1. (8. 30) Again, use the series expansion (cf.(8. 10)) involving even Chebyshev poly- nomialsT2m(cosφ) withτ=/radicalbig ν2+q2µ2, cos (νbρcosφ) cos (µaρsinφ) =∞/summationdisplay m=0/parenleftbig 2−δ0 m/parenrightbig J2m(bρτ)T2m/parenleftbig qµτ−1/parenrightbig T2m(cosφ),(8. 31) to reduce (8 .29) and (8.30) to the equivalent dual integral equations involving the Bessel function kernel of form J0(τbρ), /integraldisplay∞ 0dν/integraldisplay∞ 0dµF(ν,µ)J0(τbρ) = 1,0≤ρ<1, (8. 32) /integraldisplay∞ 0dν/integraldisplay∞ 0dµ/radicalbig ν2+µ2F(ν,µ)J0(τbρ) = 0, ρ> 1. (8. 33) When elliptic plate degenerates into circular disc ( q= 1,b= 1 ), equations identical to those obtained in the previous section are obtained. As before, we may use the substitution method to solve these disc-like equations. Themodiﬁed form of the desired solution (cf. (8. 20)) that takes into account the elliptic shape is (with τ=/radicalbig ν2+q2µ2) F(ν,µ) =/parenleftbig ν2+µ2/parenrightbig−1 2τ−1 2∞/summationdisplay k=0xkJ2k+1 2(τb) (8. 34) where the coeﬃcients xkare to be found. Insertion of this representation into Equations (8 .32) and (8.33) produces the dual equations κ(q)∞/summationdisplay k=0xk/integraldisplay∞ 0τ−1 2J0(τbρ)J2k+1 2(τb)dτ= 1,0≤ρ<1,(8. 35) ∞/summationdisplay k=0xk/integraldisplay∞ 0τ1 2J0(τbρ)J2k+1 2(τb)dτ= 0, ρ> 1, (8. 36) ©200 1 CRC Press LLC where κ(q)=/integraldisplay1 0dt/radicalbig (1−t2)[1−(1−q2)t2]=K/parenleftBig/radicalbig 1−q2/parenrightBig (8.37) isacompleteelliptici ntegralofﬁrs tkind. Asbefore ,(8.24)showsthatthesecon dEquation(8 .36)isautomatically satisﬁed;from(8 .23)onem aytransform(8 .35)to ∞/summationdisplay k=0Γ/parenleftbig k+1 2/parenrightbig Γ(k+1)xkP2k/parenleftBig/radicalbig 1−ρ2/parenrightBig =(2b)1 2/K/parenleftBig/radicalbig 1−q2/parenrightBig ,0≤ρ<1. (8.38) Thishastheclose dformsolution xk=δ0k(2b)1 2/K/parenleftBig/radicalbig 1−q2/parenrightBig (k=0,1,2,...). (8.39) Thesolutionfortheunknownfunction F(ν,µ)isdeduce dfrom(8.34)tobe F(ν,µ)=2 πsin/parenleftBig/radicalbig ν2+q2µ2b/parenrightBig /radicalbig (ν2+q2µ2)(ν2+µ2).1 K/parenleftBig/radicalbig 1−q2/parenrightBig. (8.40) Whentheellipticdisciscircular( q=1)thesolution(8 .40)coincideswith (8.19)onth eassumptionthat b=1. Wemayn owcalculatethecapacitanc eCofth eellipticplates .Atunit potential,thevalu eofCnumericallycoincide swithth etotalcharge Qac- cumulatedontheellipticplate.Thismaybecalculatedbyintegrationover thesurfac eofthesurfacechargedensi ty,whi chequalsthejumpinnormal componentofth eelectrostati cﬁeldacrossth eplate, σ(x,y)=1 4π{Ez(x,y,−0)−Ez(x,y,+0)}. Thecapacitanceisreadilyfoundtobe C=b/K/parenleftBig/radicalbig 1−q2/parenrightBig . (8.41) 8.2.1Thespherically-cur vedellipti cplate Themethodofinversionallowsustocalculatethecapacitanceofacurvedel-lipticplate.Fromth eperspectiveofinversion ,wear enaturallyledtoconsider thespherically-curvedellipti cplate,conformalwiththesurfaceofasphere, showninFigure8.3.Le tMbethece ntreofinversionofasphereofradius 2R; consider the plane tangent to this sphere at the point O /primeantipodal to M. Under inversion, the image of this tangent plane is a sphere of radius R and centre Olocated at the midpoint of the segment MO/prime.The image of an ©200 1 CRC Press LLC Figur e8.3 (a)Thesphericall yconforma lellipti cplate ;itsimag eunde rinver- sionistheellipti cdisc.(b)Thespher ewithellipti caperture ;its imag eunde rinversio nistheplanewiththeellipti cdiscremoved. ellips elyinginthetange ntplaneandcentredatO/primeissphericall yconformal; itisanelliptically-sha pedregionofthespherica lsurface .Theimageofthe tange ntplanewiththeellipti cdiscremovedisaspherica lshellwithanelliptic aperture .IntroduceaxesasshowninFigur e8.3:thez-axiscoincide swith OM,andtheusualspherica lpolars(r,θ,φ)andcylindrica lpolars(ρ,φ,z)are centredatO.Theellipti cdiscliesintheplanez=−R.Themapgivenby ρ=2Rtan1 2θ corres pondstoinversioninthespher eofradiu s2RcentredatM,followed bytheantipodalmap(r,θ,φ)/mapsto−→(r,π−θ,2π−φ);itistheimageofthe ellipti cdiscunderthismapthatisshowninFigur e8.3. The boundary of the elliptic plate is speciﬁed ρ(ϕ) =b//radicalBig 1 +κ2sin2ϕ, whereκ=q−1/radicalbig 1−q2, q=a/b; thus, the boundary of the spherically con- ©200 1 CRC Press LLC/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 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/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 /0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0 /1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1 θ θz y x x’oM o’y’ a -a b-bRa b xyz R oθbθa x’b-a o’ ay’-bMa) b) formal elliptic region is given by θ(ϕ) = 2 arctan/braceleftbiggb 2R/parenleftbig 1 +κ2sin2ϕ/parenrightbig−1 2/bracerightbigg . (8. 42) The angles θb= 2 arctan/parenleftbiggb 2R/parenrightbigg , θa= 2 arctan/parenleftBiga 2R/parenrightBig (8. 43) corresponding to the image of end-points of the semi-axes of the plane ellipse measure the angular spread of the curved plate. First consider the conformal plate charged to unit potential. According to Bouwkamp’s theorem, the problem to be solved is equivalent to the elec- trostatic problem for the grounded planar elliptic plate in the presence of aunit negative charge located at M. The free-space potential generated by this charge is ψ 0(x,y,z ) =−/braceleftBig x2+y2+ (z−2R)2/bracerightBig−1 2. (8. 44) Based upon previous results we are led to the following dual integral equations to be solved for the unknown function f, /integraldisplay∞ 0dνcosνx/integraldisplay∞ 0dµf(ν,µ) cosµx=/braceleftbig x2+y2+ 4R2/bracerightbig−1 2, (8. 45) /integraldisplay∞ 0dνcosνx/integraldisplay∞ 0dµ/radicalbig ν2+µ2f(ν,µ) cosµx= 0. (8. 46) The ﬁrst equation holds for points ( x,y) lying inside the disc, whilst the second holds for those points outside. Substituting (8. 28), we obtain /integraldisplay∞ 0dνcos (νbρcosφ)/integraldisplay∞ 0dµf(ν,µ) cos (µaρsinφ) =b−1/braceleftbig ρ2+γ2−k2ρ2sin2φ/bracerightbig−1 2, 0≤ρ<1,(8. 47) /integraldisplay∞ 0dνcos (νbρcosφ)/integraldisplay∞ 0dµ/radicalbig ν2+µ2f(ν,µ) cos (µaρsinφ) = 0, ρ> 1, (8. 48) whereγ= 2R/bandk=/radicalbig 1−q2; this holds for 0 ≤φ<1 2π. Expand the right-hand side of (8. 47) in a Chebyshev series /braceleftbig ρ2+γ2−k2ρ2sin2φ/bracerightbig−1 2=∞/summationdisplay m=0(2−δ0m)α2mT2m(cosφ), (8. 49) where α2m=α2m(ρ) =2 π/integraldisplayπ 2 0T2m(cosφ)dφ/radicalbig ρ2+γ2−k2ρ2sin2φ. (8. 50) ©200 1 CRC Press LLC In particular, α0=α0(ρ) =2 π/radicalbig ρ2+γ2K/parenleftBigg kρ/radicalbig ρ2+γ2/parenrightBigg , (8. 51) whereKdenotes the complete elliptic integral of ﬁrst kind. It is evident that (8. 47) and (8. 48) imply that /integraldisplay∞ 0dν/integraldisplay∞ 0dµf 2m(ν,µ)J2m(/radicalbig ν2+q2µ2bρ) =b−1α2m(ρ), ρ< 1,(8. 52) /integraldisplay∞ 0dν/integraldisplay∞ 0dµ/radicalbig ν2+µ2f2m(ν,µ)J2m(/radicalbig ν2+q2µ2bρ) = 0, ρ> 1,(8. 53) wherem= 0,1,2,..., and f2m(ν,µ) =T2m/parenleftBigg qµ/radicalbig ν2+q2µ2/parenrightBigg f(ν,µ). (8. 54) To ﬁnd the solution use the extended form of the representation (8. 34) f2m(ν,µ) =/parenleftbig ν2+µ2/parenrightbig−1 2(ν2+q2µ2)−1 4∞/summationdisplay n=0xm kJ2k+2m+1 2(/radicalbig ν2+q2µ2b). (8. 55) Its substitution in (8. 52) and (8. 53) produces κ(q)∞/summationdisplay k=0xm k/integraldisplay∞ 0τ−1 2J2k+2m+1 2(τb)J2m(τbρ)dτ=b−1α2m,0≤ρ<1, (8. 56) ∞/summationdisplay k=0xm k/integraldisplay∞ 0τ1 2J2k+2m+1 2(τb)J2m(τbρ)dτ= 0, ρ> 1. (8. 57) We employ the generalisation of the integrals given in (8. 23) and (8. 24), /integraldisplay∞ 0τ1 2J2k+2m+1 2(τb)J2m(τbρ)dτ =2−2m+1 2 b3 2Γ (k+ 1) Γ/parenleftbig k+ 2m+1 2/parenrightbigP2m 2k+2m/parenleftBig/radicalbig 1−ρ2/parenrightBig /radicalbig 1−ρ2H(1−ρ),(8. 58) /integraldisplay∞ 0τ−1 2J2k+2m+1 2(τb)J2m(τbρ)dτ =2−2m−1 2 b1 2Γ/parenleftbig k+1 2/parenrightbig Γ (k+ 2m+ 1)P2m 2k+2m/parenleftBig/radicalbig 1−ρ2/parenrightBig ,0≤ρ<1,(8. 59) ©200 1 CRC Press LLC to deduce ∞/summationdisplay k=0Γ/parenleftbig k+1 2/parenrightbig Γ (k+ 2m+ 1)xm kP2m 2¯k+2m/parenleftBig/radicalbig 1−ρ2/parenrightBig =22m+1 2 b1 2κ(q)α2m(ρ),0≤ρ<1, (8. 60) wherem= 0,1,2,.... The solution of this equation immediately follows by exploiting the orthogonality property of the associated Legendre functions on (0,1): /integraldisplay1 0ρ/radicalbig 1−ρ2P2m 2k+2m/parenleftBig/radicalbig 1−ρ2/parenrightBig P2m 2s+2m/parenleftBig/radicalbig 1−ρ2/parenrightBig dρ =1 4k+ 4m+ 1Γ (2k+ 4m+ 1) Γ (2k+ 1)δks.(8. 61) As a result we obtain xm s=/parenleftbigg2 b/parenrightbigg1 2 2−2m(4s+ 4m+ 1) Γ (s+ 1) Γ/parenleftbig s+ 2m+1 2/parenrightbig Γ/parenleftbig s+1 2/parenrightbigβsm κ(q), (8. 62) where βsm=/integraldisplay1 0ρ/radicalbig 1−ρ2α2m(ρ)P2m 2s+2m/parenleftBig/radicalbig 1−ρ2/parenrightBig dρ. (8. 63) We may now calculate the capacitance Cof the spherically conforming elliptic plate. By Bouwkamp’s theorem, it is proportional to the value of the induced potential at the centre of inversion M: C= 4R2ψ(0,0,2R). (8. 64) It is readily seen that the calculation of Conly requires a knowledge of the functionf0(ν,µ) =f(ν,µ).Let us now demonstrate the solution of Equations (8. 52) and (8. 53) (with m= 0) by the Abel integral transform method. Based on the results at the beginning of this section (see also (8. 55)), let usseek the unknown function fin the form f(ν,µ) =/parenleftbig ν 2+µ2/parenrightbig−1 2F(/radicalbig ν2+q2µ2). (8. 65) After some evident manipulation, we obtain the dual integral equations /integraldisplay∞ 0F(τ)J0(τρb)dτ=2 πb1/radicalbig ρ2+γ21 K/parenleftBig/radicalbig 1−q2/parenrightBigK/parenleftBigg/radicalbig 1−q2ρ/radicalbig ρ2+γ2/parenrightBigg , 0≤ρ<1,(8. 66) /integraldisplay∞ 0τF(τ)J0(τρb)dτ= 0, ρ> 1. (8. 67) ©200 1 CRC Press LLC Useth emethoddescribe dinSection2.7totransformthes edualequationsto the Fourier cosine form /integraldisplay∞ 0F(τ) cosτρb dτ =b−1γ K/parenleftBig/radicalbig 1−q2/parenrightBig/parenleftbig γ2+ρ2/parenrightbig−1 2/parenleftbig γ2+q2ρ2/parenrightbig−1 2H(1−ρ), (8. 68) and invert this expression to obtain F(τ) =2 πb−1γ K/parenleftBig/radicalbig 1−q2/parenrightBig/integraldisplay1 0cosτρb/radicalbig (γ2+ρ2) (γ2+q2ρ2)dρ. (8. 69) According to (8. 64) the capacitance of the spherically-conforming elliptic plate is C= 4R2/integraldisplay1 0/braceleftbig/parenleftbig 1−t2/parenrightbig/bracketleftbig 1−/parenleftbig 1−q2/parenrightbig t2/bracketrightbig/bracerightbig−1 2× /integraldisplay∞ 0F(τ)e−τ√ 1−(1−q2)t22R/qdτdt. (8. 70) Remarkably, substitution of (8. 69) into (8. 70) produces the closed form expression C=2R K/parenleftBig/radicalbig 1−q2/parenrightBig/braceleftbiggarctanγ−1−qarctanqγ−1 1−q2/bracerightbigg , (8. 71) which may be written in terms of the angles θa,θb(deﬁned by (8. 43)) as C=R K/parenleftBig/radicalbig 1−q2/parenrightBig/braceleftbiggθb−qθa 1−q2/bracerightbigg . (8. 72) When the elliptic plate degenerates to a circular disc ( q→1), the conforming plate becomes a spherical cap; since K(0) =1 2πand lim q→1arctanγ−1−qarctanqγ−1 1−q2=1 2/braceleftBig arctanγ−1+γ/parenleftbig 1 +γ2/parenrightbig−1/bracerightBig =1 4{θb+ sinθb}(8. 73) the expression for its capacitance reduces to the well-known value previously calculated for the spherical cap, namely π−1(θb+ sinθb). This completes our discussion of the capacitance of the spherically conform- ing elliptic plate. The complementary structure – the spherical shell with anelliptic aperture – may be analysed in a similar fashion. ©200 1 CRC Press LLC Figur e8.4 Thepolygona lplateandcircumscribin gcircle. 8.3Polygona lplates Incontrasttotheplate swithsmoothboundarie sconsidere dinprevious sections ,thissectio nexamine spolygona lplates ,particularl yregula rpolygons ofNequalsides(N=3,4,...).AsshowninFigur e8.4,theanglesubtended by each side at the centre Oof the polygon is 2 α= 2π/N. If the circle circumscribing the polygon has radius a, the diﬀerence in length between an edgeABof the polygon and the circular arc ABof the circumscribing circle is a(2π/N−2 sinπ/N) ; asn→ ∞,this diﬀerence is1 3π3/N3+O(N−5),and the circle approximates the polygon in some sense. When the plate is charged, symmetry implies that we may concentrate on the right-angled triangular sectorOAC , where the angle /hatwideOAC =α. The potential on the charged circular plate S0=/braceleftbig (x,y,0) :x2+y2<a2/bracerightbig is determined by the dual equations of the form (see (8. 16)) /integraldisplay∞ 0dνcosνx/integraldisplay∞ 0dµf(µ,ν) cosµy= 1,(x,y)∈S0, (8. 74) /integraldisplay∞ 0dνcosνx/integraldisplay∞ 0dµ/radicalbig ν2+µ2f(µ,ν) cosµy= 0,(x,y)/∈S0. (8. 75) ©200 1 CRC Press LLCy x o αα BaA cc ’ϕ ϕ=0 Thesedualintegralequation swer esolve dinSection8.1.Thesameequations hold for the polygon charged to unit potential, except the region S0is diﬀer- ently deﬁned. It is suﬃcient to consider the triangular region OAC and the associated unbounded sector deﬁned by angle α. In the limit when N→ ∞,this sector degenerates to a half-line or ray. If we consider the ray y= 0, the equations (8. 74), (8. 75) are /integraldisplay∞ 0dµ/integraldisplay∞ 0dνf(µ,ν) cosνx= 1,0<x<a, (8. 76) /integraldisplay∞ 0dµ/integraldisplay∞ 0dν/radicalbig ν2+µ2f(µ,ν) cosνx= 0, x>a. (8. 77) The substitution τ=/radicalbig ν2+µ2leads to the readily solvable equations for the potential distribution on the circular disc, /integraldisplay∞ 0τf(τ)J0(τx)dτ=2 π,0<x<a, (8. 78) /integraldisplay∞ 0τ2f(τ)J0(τx)dτ= 0, x>a. (8. 79) We shall solve the potential problem by transforming the dual equations to a form that may be recognised as a perturbation of the equations describingthe circular disc. Settingy=xtanφ,we concentrate on the sector deﬁned by φ∈(0,π/N ). The dual equations corresponding to (8. 74) and (8. 75) are /integraldisplay ∞ 0dνcosνx/integraldisplay∞ 0dµf(µ,ν) cos (µxtanφ) = 1,(8. 80) /integraldisplay∞ 0dνcosνx/integraldisplay∞ 0dµ/radicalbig ν2+µ2f(µ,ν) cos (µxtanφ) = 0,(8. 81) where the ﬁrst equation hods for x∈(0,acos (π/N)),and the second for x∈(acos (π/N),∞) respectively. The substitutions ρ=xsec (π/N), u= tanφcot (π/N), (8. 82) transform these dual equations to /integraldisplay∞ 0dνcos/parenleftBig νρcosπ N/parenrightBig/integraldisplay∞ 0dµf(µ,ν) cos/parenleftBig µρusinπ N/parenrightBig = 1, (8. 83)/integraldisplay∞ 0dνcos/parenleftBig νρcosπ N/parenrightBig/integraldisplay∞ 0dµ/radicalbig ν2+µ2f(µ,ν) cos/parenleftBig µρusinπ N/parenrightBig = 0, (8. 84) ©200 1 CRC Press LLC wheretheﬁrstequationhold sforρ∈(0,a),u∈(0,1)andthesecon dfor ρ∈(a,∞),u∈(0,1).ArguingasinSection8.1,thedependenceu ponuin these equations can be eliminated by transformation to the form /integraldisplay∞ 0dνcos/parenleftBig νρcosπ N/parenrightBig/integraldisplay∞ 0dµf(µ,ν)J0/parenleftBig µρsinπ N/parenrightBig = 1, (8. 85)/integraldisplay∞ 0dνcos/parenleftBig νρcosπ N/parenrightBig/integraldisplay∞ 0dµ/radicalbig ν2+µ2f(µ,ν)J0/parenleftBig µρsinπ N/parenrightBig = 0, (8. 86) holding for ρ∈(0,a) andρ∈(a,∞),respectively. It should be observed that whenN→ ∞,Equations (8. 85) and (8. 86) degenerate to (8. 76) and (8. 77). When the plate is a circular or elliptic disc, the dual equations analogous to (8. 85) and (8. 86) have particularly simple solutions of the form f(µ,ν) = f(τ) (whereτ=/radicalbig ν2+µ2for the circular disc, and τ=/radicalbig ν2+q2µ2for the elliptic disc). It is not obvious a priori that the solution f(µ,ν) to (8. 85) and (8. 86) has a solution of a similarly simple form. However, it turns out that the form is exactly the same as that for the circular disc; thus, we shallassume f(µ,ν) =f(τ),whereτ=/radicalbig ν2+µ2, (8. 87) and justify this assumption retrospectively by showing that the solution soconstructed satisﬁes all equations and associated conditions. With this as-sumption, the dual equations become /integraldisplay ∞ 0F(τ)SN(τρ)dτ= 1, ρ∈(0,a), (8. 88) /integraldisplay∞ 0τF(τ)SN(τρ)dτ= 0, ρ∈(a,∞), (8. 89) whereF(τ) =τf(τ) and the kernel SNis deﬁned by SN(τρ) =/integraldisplayτ 0J0/parenleftbig√ τ2−ν2ρsinπ N/parenrightbig √ τ2−ν2cos/parenleftBig νρcosπ N/parenrightBig dτ =/integraldisplayτ 0cos/parenleftbig√ τ2−ν2ρcosπ N/parenrightbig √ τ2−ν2J0/parenleftBig νρsinπ N/parenrightBig dτ =π 2J0/parenleftBig τρcos2π 2N/parenrightBig J0/parenleftBig τρsin2π 2N/parenrightBig .(8. 90) WhenN→ ∞,the kernel becomes S∞(τρ) = lim N→∞SN(τρ) =π 2J0(τρ), (8. 91) which is identical with that encountered for the circular disc. ©200 1 CRC Press LLC This construction justiﬁes our assumption of the form (8. 87) for f. We may therefore seek the solution to the dual equations in the form F(τ) =/integraldisplaya 0G(τ) cos/parenleftBig τtcosπ N/parenrightBig dt (8. 92) where both the function Gand its derivative G/primeare continuous on (0 ,a). Integrating by parts, Fis representable as F(τ) = secπ N/braceleftBigg G(a)sin/parenleftbig aτcosπ N/parenrightbig τ−1 τ/integraldisplaya 0G/prime(τ) sin/parenleftBig τtcosπ N/parenrightBig dt/bracerightBigg . (8. 93) Now substitute (8. 93) into (8. 89) and invert the order of integration. Then whenρ>a, G(a)/integraldisplay∞ 0SN(τρ) sin/parenleftBig aτcosπ N/parenrightBig dτ− /integraldisplaya 0G/prime(t)/braceleftBig SN(τρ) sin/parenleftBig tτcosπ N/parenrightBig dτ/bracerightBig dt= 0.(8. 94) However, it is well known (see [19]) that /integraldisplay∞ 0Jν(ax)Jν(bx) sinxy dx = 0, 0<y<b −a, (8. 95) whenb>a, Reν >−1,so that the equation (8. 94) holds identically. Following the basic idea of regularisation, we split the kernel SNas a sum of its limiting value S∞and a correction term and analytically invert that part of the equation containing the limiting kernel contribution, corresponding to the circular disc problem. This is most naturally done in the present context by using the result derived from the addition theorem for Bessel functions[14], J 0/parenleftBig τρcos2π 2N/parenrightBig J0/parenleftBig τρsin2π 2N/parenrightBig =J0(τρ)−2∞/summationdisplay n=1(−1)nJn/parenleftBig τρcos2π 2N/parenrightBig Jn/parenleftBig τρsin2π 2N/parenrightBig .(8. 96) We may now construct the representation of the function to be determined. First expand Gin a series with Gegenbauer polynomials C(1 2) 2k=P2k, G(t) =∞/summationdisplay k=1bkC(1 2) 2k(t/a). (8. 97) Substitute this expression in (8. 92), invert the order of summation and integration and obtain F(τ) =∞/summationdisplay k=1bk/integraldisplaya 0cos/parenleftBig τtcosπ N/parenrightBig C(1 2) 2k(t/a)dt. (8. 98) ©200 1 CRC Press LLC Using the tabulated integral [14] (Vol. 1) /integraldisplaya 0cos/parenleftBig τtcosπ N/parenrightBig C(1 2) 2k(t/a)dt = (−1)k/parenleftBigπa 2secπ N/parenrightBig1 2τ−1 2J2k+1 2/parenleftBig τacosπ N/parenrightBig ,(8. 99) we deduce that F(τ) =τ−1 2∞/summationdisplay k=1b∗ kJ2k+1 2/parenleftBig τacosπ N/parenrightBig , (8. 100) where b∗ k= (−1)k/parenleftBigπa 2secπ N/parenrightBig1 2bk. (8. 101) Substitute (8. 100) into (8. 88) and change the order of integration and summation to obtain ∞/summationdisplay k=1b∗ k/integraldisplay∞ 0τ−1 2SN(τρ)J2k+1 2/parenleftBig τacosπ N/parenrightBig dτ= 1, ρ∈(0,a).(8. 102) We recall that Equation (8. 89) is satisﬁed automatically with the repre- sentation (8. 92) or its equivalent form (8. 100). After some manipulation, we deduce from (8. 100) that ∞/summationdisplay k=0b∗ kΓ/parenleftbig k+1 2/parenrightbig Γ (k+ 1)P2k/parenleftBig/radicalbig 1−ρ2/a2/parenrightBig =2 π/parenleftBig 2acosπ N/parenrightBig1 2+∞/summationdisplay k=0b∗ kΓ/parenleftbig k+1 2/parenrightbig Γ (k+ 1)Fk(ρ), ρ∈(0,a),(8. 103) where Fk(ρ) =P2k/parenleftBig/radicalbig 1−ρ2/a2/parenrightBig −/integraldisplayπ 0℘k(ρ,x)dx, (8. 104) ℘k(ρ,x) =/braceleftBigg π−1P2k/parenleftBig/radicalbig 1−ρ2/ρ2c/parenrightBig , ρ<ρ c, 2π−1 2arcsin (ρ/ρc) Γ/parenleftbig k+1 2/parenrightbig /Γ (k+ 1), ρ>ρ c,(8.105) and the value of ρcis deﬁned by the relation a ρc= secπ N/parenleftBig 1−sin2π Ncos2π N/parenrightBig1 2. (8. 106) It is evident that as N→ ∞, Fk(ρ)→0. Apply the usual principle of orthogonality of Legendre polynomials on the interval [0,1] to obtain the i.s.l.a.e. of the second kind, xs−∞/summationdisplay k=0γksxk=2 πδ0s, (8. 107) ©200 1 CRC Press LLC fors=0,1,2,...,where b∗ k=/parenleftBig 2acosπ N/parenrightBig1 2Γ(k+1) Γ/parenleftbig k+1 2/parenrightbig(4k+1)1 2xk, (8.108) and γks=[(4k+1)(4s+1)]1 2/integraldisplay1 0t√ 1−t2Fk(t)P2s/parenleftBig/radicalbig 1−t2/parenrightBig dt. (8.109) Thesolutionofthissystemofequations {xk}∞ k=0issoughtin l2.The computationofth eintegral sdeﬁnin gthematrixelementsisstraightfor ward. Furthermore,as N→∞,Fk(ρ)→0andestimatesofth ediﬀerencebetween thepotentialdistributionforacirculardiscan dapolygonaldiscwithmany vertice s(N/greatermuch1)arereadilyderivedfrom(8.107). 8.4Theﬁnit estrip InSection7.2weexamine dthepotentialass ociate dwithchargedinﬁnitely longthinstrips.Althoughthistwo-dimensionalproblemhasitsownintrinsic interest,itisworthexaminingth emorephysicallyrealisticstructureofa ﬁnitelylongstrip.Considertheﬂatstripofwidth2 aandlength2 b>2a lyingintheplane z=0asshowninFigure8.5.Thece ntrelie sattheorigin and the edges are aligned with the xandyaxes. Suppose the strip is charged to unit potential. The mixed boundary conditions satisﬁed by the electrostaticpotentialψare ψ(x,y,+0) =ψ(x,y,−0) = 1,|x| ≤a,|y| ≤b, (8. 110) and by its normal derivative are ∂ ∂zψ(x,y,+0) =∂ ∂zψ(x,y,+0),|x|>aor|y|>b. (8. 111) The symmetry of the structure leads to the familiar form (8. 16) for thesolution, and enforcement of the mixed boundary conditions leads to dual integral equations for the unknown function F=F(ν,µ), /integraldisplay ∞ 0dνcosνbx/prime/integraldisplay∞ 0dµF(ν,µ) cos (µay/prime) = 1,|x/prime| ≤1,|y/prime| ≤1,(8. 112) /integraldisplay∞ 0dνcosνbx/prime/integraldisplay∞ 0dµ/radicalbig ν2+µ2F(ν,µ) cos (µay/prime) = 0,|x/prime|>1 or|y/prime|>1 (8. 113) ©200 1 CRC Press LLC Figure 8.5 The ﬁnite strip. wherex/prime=x/a, y/prime=y/b. The distinctive feature of these equations is the apparent lack of coupling between the rescaled variables x/primeandy/prime.This dictates a special choice for the form of the solution to be found by the substitution method. In order to satisfy (8. 113) automatically, it is suﬃcient to represent the unknownfunctionFby an expansion in Bessel functions of even order, F(ν,µ) =/parenleftbig ν 2+µ2/parenrightbig−1 2∞/summationdisplay n=0∞/summationdisplay m=0xnmJ2n(νb)J2m(µa), (8. 114) where the coeﬃcients xnmare to be determined. Substitution of this form in (8. 113) leads to ∞/summationdisplay n=0∞/summationdisplay m=0xnm/integraldisplay∞ 0dνcosνbx/primeJ2n(νb)/integraldisplay∞ 0dµcos (µay/prime)J2m(µa) = 0,(8. 115) when|x/prime|>1 or|y/prime|>1.The product of integrals occurring in (8. 115) vanish because [19] /integraldisplay∞ 0J2n(αx) cosxydx = (−1)n/parenleftbig α2−y2/parenrightbig−1 2T2n(y/α)H/parenleftbig α2−y2/parenrightbig .(8. 116) ©200 1 CRC Press LLCz xa -a o-b bΨ= 1o y Moreover, it is apparent from (8. 116) that the behaviour of the surface charge densityσ(x/prime,y/prime) near the edges will be in accord with physical expectation, namely σ(x/prime,y/prime)/revsimilarσ0/parenleftbig b2−x2/parenrightbig−1 2/parenleftbig a2−y2/parenrightbig−1 2(σ0constant). Now substitute (8. 114) into (8. 112). Using the expansions cos (νbx/prime) =∞/summationdisplay s=0(−1)s(2−δ0s)J2s(νb)T2s(x/prime), cos (µay/prime) =∞/summationdisplay p=0(−1)p(2−δ0p)J2p(µa)T2p(y/prime), and the orthogonality of the Chebyshev polynomials on [0 ,1], we obtain the i.s.l.a.e. for the unknowns xnm(n,m = 0,1,2,...), ∞/summationdisplay n=0∞/summationdisplay m=0xnmRnmsp =δ0sδ0p, (8. 117) wheres,p= 0,1,2,..., and matrix elements Rnmsp are given by (−1)s+p/integraldisplay∞ 0/integraldisplay∞ 0dνdµ/parenleftbig ν2+µ2/parenrightbig−1 2J2n(νb)J2m(µa)J2s(νb)J2p(µa) =(−1)s+p b/integraldisplay∞ 0duJ 2n(u)J2s(u)/integraldisplay∞ 0dv/parenleftbig u2+v2/parenrightbig−1 2J2m(qv)J2p(qv), (8. 118) withq=a/b. This reduction to the i.s.l.a.e. (8. 117) is a very formal proce- dure. The representation (8. 118) of the matrix elements Rnmsp in terms of slowly convergent iterated integrals makes numerical procedures problematic. Let us transform (8. 118), where for convenience we will set b= 1. Making use of the representation for the product of Bessel functions [14] J2n(u)J2s(u) =2 π/integraldisplayπ 2 0J2n+2s(2ucosθ) cos [(2s−2n)θ]dθ, (8. 119) valid when Re ( ν+µ)>−1,and the tabulated integral [14] /integraldisplay∞ 0Jν(cx)dx√ x2+z2=I1 2ν/parenleftBigcz 2/parenrightBig K1 2ν/parenleftBigcz 2/parenrightBig , (8. 120) valid when c>0,Rez >0,Reν >−1,the expression for the matrix element Rnmsp becomes Rnmsp =2 π(−1)s+p/integraldisplayπ 2 0cos (2n−2s)φ× /braceleftbigg/integraldisplay∞ 0J2m(qv)J2p(qv)In+s(vcosφ)Kn+s(vcosφ)dv/bracerightbigg dφ. (8. 121) ©200 1 CRC Press LLC Using the Mellin transform one may represent the product of modiﬁed Bessel functions occurring in (8. 121) in the form (see [61]) In+s(vcosφ)Kn+s(vcosφ) = =1 8π3 2i/integraldisplayc+i∞ c−i∞Γ/parenleftbig n+s+t 2/parenrightbig Γ/parenleftbigt 2/parenrightbig Γ/parenleftbig1 2−t 2/parenrightbig Γ/parenleftbig n+s+ 1−t 2/parenrightbig cos−tφv−tdt, (8. 122) where 0<c< 1.After substitution of (8. 122) into (8. 121) and some obvious rearrangement, the expression for the matrix element takes the form Rnmsp =(−1)s+p 4π5 2i/integraldisplayc+i∞ c−i∞Γ/parenleftbig n+s+t 2/parenrightbig Γ/parenleftbigt 2/parenrightbig Γ/parenleftbig1 2−t 2/parenrightbig Γ/parenleftbig n+s+ 1−t 2/parenrightbigAns(t)Bmp(t)dt (8. 123) where Ans(t) =/integraldisplayπ 2 0cos−tφcos (2n−2s)φdφ, (8. 124) Bmp(t) =/integraldisplay∞ 0v−tJ2m(qv)J2p(qv)dv. (8. 125) Both integrals occurring in (8. 124) and (8. 125) are tabulated in [19], and so Ans(t) =√π 2Γ/parenleftbig1 2−t 2/parenrightbig Γ/parenleftbig 1−t 2/parenrightbig Γ/parenleftbig s−n+ 1−t 2/parenrightbig Γ/parenleftbig n−s+ 1−t 2/parenrightbig, (8. 126) Bmp(t) =qt−1 2√πΓ/parenleftbigt 2/parenrightbig Γ/parenleftbig1 2+t 2/parenrightbig Γ/parenleftbig p+m+1 2−t 2/parenrightbig Γ/parenleftbig m−p+1 2+t 2/parenrightbig Γ/parenleftbig p+m+1 2+t 2/parenrightbig Γ/parenleftbig p−m+1 2+t 2/parenrightbig. (8. 127) Insert (8. 126) and (8. 127) into (8. 123), make the substitution t= 2r+ 1, and replace rbyt, to obtain Rnmsp =(−1)s+p 4π3 21 2πi/integraldisplay LΓ2/parenleftbig1 2+t/parenrightbig Γ2(−t) Γ/parenleftbig1 2−t/parenrightbig Γ (1 +t) Γ/parenleftbig n+s+1 2−t/parenrightbig Γ/parenleftbig s−n+1 2−t/parenrightbig× Γ/parenleftbig n+s+1 2+t/parenrightbig Γ (p+m−t) Γ/parenleftbig n−s+1 2−t/parenrightbig Γ (m−p+ 1 +t)× 1 Γ (p+m+ 1 +t) Γ (p−m+ 1 +t)q2tdt(8. 128) where the contour Lruns from −i∞to +i∞, intersecting the real axis at a pointt0satisfying the inequality −1 2< t 0<0. It is evident that all the poles of Γ ( −t) and Γ (λ−t) lie to the right of L, whereas all the poles of Γ (1 +t),Γ/parenleftbig1 2+t/parenrightbig and Γ (µ+t) lie to the left of L. We may express the ©200 1 CRC Press LLC contourintegra lintermsofMeijer’ sG-function ,asdeﬁne din[14],via Rnmsp= (−1)s+p 4π3 2G4,4 7,7/parenleftbigg q2|1 2,1 2,0,−n−s+1 2,n+s+1 2,s−n+1 2,n−s+1 2 0,0,1 2,p+m,p−m,−p−m,m−p/parenrightbigg . (8.129) Whens=nandp=m,simpl eidentitiessatisﬁe dbyMeijer’ sG-function showthatthe“diagonal ”matri xeleme ntsRnmn maregivenby Rnmn m=(−1)n+m 4π3 2G3,3 5,5/parenleftbigg q2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 2,1 2,−2n+1 2,2n+1 2,1 2 0,0,2m, 0,−2m/parenrightbigg .(8.130) Usingthewell-kn ownrelation sfortheGamm afunctio n(seeAppendix ,(B. 3)) Γ (−t) Γ (1 +t) =−π sin (πt), Γ (λ−t) Γ (−λ+ 1 +t) =−(−1)λπ sin (πt), Γ/parenleftbigg µ+1 2−t/parenrightbigg Γ/parenleftbigg −µ+1 2+t/parenrightbigg = (−1)µπ cos (πt), we may derive the expression Rnmsp =−(−1)p+m 4√π1 2πi/integraldisplay Lcos2(πt) sin3(πt)Γ/parenleftbig1 2+t/parenrightbig Γ (1 +t)Γ/parenleftbig n+s+1 2+t/parenrightbig Γ (−p+m+ 1 +t)× Γ/parenleftbig −n−s+1 2+t/parenrightbig Γ/parenleftbig −s+n+1 2+t/parenrightbig Γ/parenleftbig s−n+1 2+t/parenrightbig Γ (m−p+ 1 +t) Γ (p+m+ 1 +t) Γ (p−m+ 1 +t)q2tdt. (8. 131) Evaluation of the contour is thus reduced to the evaluation of residues at the polest= 0,1,2,.... After some manipulation, this yields Rnmsp =(−1)µ 8π∞/summationdisplay k=0/parenleftbig1 2/parenrightbig k/parenleftbig κ+1 2/parenrightbig k/parenleftbig −κ+1 2/parenrightbig k/parenleftbig λ+1 2/parenrightbig k/parenleftbig −λ+1 2/parenrightbig k k! (µ+k)! (ν+k)!× q2kNk κλµν (8. 132) whereκ=n+s,λ=s−n,µ=p+m,ν =p−m,and the coeﬃcients Nk κλµν ©200 1 CRC Press LLC are deﬁned as follows. When µ≤kandν≤k, Nk κλµν =(−1)k (−µ+k)! (−ν+k)!/braceleftbig π2+ψ/prime(1 +k) + ψ/prime(µ+ 1 +k) +ψ/prime(−µ+ 1 +k) + ψ/prime(ν+ 1 +k) +ψ/prime(−ν+ 1 +k)− ψ/prime/parenleftbigg1 2+k/parenrightbigg −ψ/prime/parenleftbigg κ+1 2+k/parenrightbigg −ψ/prime/parenleftbigg −κ+1 2+k/parenrightbigg − ψ/prime/parenleftbigg λ+1 2+k/parenrightbigg −ψ/prime/parenleftbigg −λ+1 2+k/parenrightbigg − (ψ/parenleftbigg1 2+k/parenrightbigg +ψ/parenleftbigg κ+1 2+k/parenrightbigg +ψ/parenleftbigg −κ+1 2+k/parenrightbigg + ψ/parenleftbigg λ+1 2+k/parenrightbigg +ψ/parenleftbigg −λ+1 2+k/parenrightbigg + 2 logq−ψ(1 +k)− ψ(µ+ 1 +k)−ψ(−µ+ 1 +k)−ψ(ν+ 1 +k)−ψ(−ν+ 1 +k))2/bracerightbig . (8. 133) Whenµ > k andν≤k, orµ≤kandν > k , it is necessary to remove the indeterminacy which appears in this formula arising from the product of zero and inﬁnite terms by use of the formulae ψ(x) Γ (x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle x=−j=−d dx/bracketleftbigg1 Γ (x)/bracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle x=−j= (−1)j−1Γ (j+ 1), (8. 134) ψ/prime(x)−ψ2(x) Γ (x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle x=−j= 2 (−1)jΓ (j+ 1)ψ(j+ 1) (8. 135) wherej= 0,1,2,.... Thus when µ>k andν≤k, the expression becomes Nk κλµν = 2 (−1)µΓ (µ−k) Γ (−ν+ 1 +k)/braceleftbig Fk κλµν−ψ(µ−k)−ψ(−ν+ 1 +k)/bracerightbig (8. 136) where Fk κλµν =ψ/parenleftbigg1 2+k/parenrightbigg +ψ/parenleftbigg κ+1 2+k/parenrightbigg +ψ/parenleftbigg −κ+1 2+k/parenrightbigg + ψ/parenleftbigg λ+1 2+k/parenrightbigg +ψ/parenleftbigg −λ+1 2+k/parenrightbigg + 2 logq− ψ(1 +k)−ψ(µ+ 1 +k)−ψ(ν+ 1 +k) ; (8. 137) whenµ≤kandν >k , the expression becomes Nk κλµν = 2 (−1)νΓ (ν−k) Γ (−µ+ 1 +k)/braceleftbig Fk κλµν−ψ(ν−k)−ψ(−µ+ 1 +k)/bracerightbig . ©200 1 CRC Press LLC Finallywhe nµ>k andν>k ,Formulae(8.135)maybeusedtosh owthat Nk κλµν =2(−1)k+1Γ(ν−k)Γ(µ−k). Fromthisﬁnalformi tmaybeshownthatthediagonaltermsofthematrix elementsRnmsp dominatesothatth esyste m(8.117)issatisfactoryfor computation.Itmaybeveriﬁedthatfornarrowstrip satleas t(q/lessmuch1), thisi.s.l.a.e .isnon-singular,andananalyti csolutioncanbede veloped .In thegeneralcase(0 <q< 1)numericaltechniquesm aybeemployed .This completesou rregularisationofth edualintegralequationsass ociate dwiththe ﬁnitestrip. 8.5Coupledcharge dconductors:thesphericalcapand circulardisc InSection2.9,webrieﬂydescribedtechnique sforcalculatingthepotential distributionsurroundingcouple dcharge dconductorswithcompone ntsthat arepartsofcoordinatesurface sbelongin gto diﬀerent coordinatesystems. Oneofthesimples texample sisthecombinationofasphericalcapan dcircular disc. Supposethecirculardis cofradius aislocate dintheplane z=0with centreattheorigin O;thesphericalcapals ohasitsce ntreatO,subtend san angleθ0≤1 2πatO,andhasradius b>a ;letq=a/b.(SeeFigure8.6.)Both disc and cap are charged to unit potential. Following the usual principle of superposition, the total potential Umay be expressed as the sum of two contributions U=Uc+Ud, (8. 138) where the cap contribution may be represented in the form Uc=∞/summationdisplay n=0xnPn(cosθ)/braceleftbigg (r/b)n, r<b, (r/b)−n−1,r>b,(8.139) whilst the disc contribution may be represented as Ud=/integraldisplay∞ 0G(ν)J0(νρ)e−ν|z|dν. (8. 140) The unknown coeﬃcients {xn}∞ n=0and function Gare to be found. ©200 1 CRC Press LLC Figure 8.6 The coupled disc and spherical cap. The obvious boundary conditions to be enforced are U(b,θ) = 1, θ∈(0,θ0), (8. 141) /bracketleftbigg∂ ∂rU(r,θ)/bracketrightbiggr=b+0 r=b−0= 0, θ∈(θ0,π), (8. 142) U(ρ,0) = 1, ρ∈(0,a), (8. 143) /bracketleftbigg∂ ∂zU(ρ,z)/bracketrightbiggr=b+0 r=b−0= 0, ρ∈(a,∞), (8. 144) where (8. 141) and (8. 142) have been expressed in terms of the standard spherical coordinate system ( r,θ,φ ) centred at O,whereas (8. 143) and (8. 144) have been expressed in terms of the standard cylindrical coordinate sys- tem (ρ,φ,z ) centred at O(so thatρ=rsinθ,z=rcosθ). Enforcement of the boundary conditions (8. 141)–(8. 144) leads to the integro-series equations for the unknowns ∞/summationdisplay n=0xnPn(cosθ) = 1−/integraldisplay∞ 0G(ν)J0(νbsinθ)e−νb|cosθ|dν, θ∈(0,θ0),(8. 145) ∞/summationdisplay n=0(2n+ 1)xnPn(cosθ) = 0, θ ∈(θ0,π), (8. 146) /integraldisplay∞ 0G(ν)J0(νρ)dν= 1−∞/summationdisplay n=0x2nP2n(0)(ρ/b)2n, ρ∈(0,a), (8. 147) ©200 1 CRC Press LLC /integraldisplay∞ 0νG(ν)J0(νρ)dν=0,ρ ∈(a,∞). (8.148) Inderiving(8.147) ,theproperty P2n+1(0)=0wasused.Usingthemeth od describedinSection2.6,andnotin gthevalue P2n(0)=( −1)n(2n−1)!! (2n)!!, wemaytransform(8.147)and(8 .148)toth eform /integraldisplay∞ 0G(ν)cosνρdν =/braceleftBigg 1−∞/summationdisplay n=0(−1)nx2n(ρ/b)2n/bracerightBigg H(a−ρ).(8.149) TheapplicationofaninversecosineFouriertransfor mto(8.149)produces ourﬁrstintegro-seriesequationinalgebrai cform, G(ν)+∞/summationdisplay n=0x2nRn(ν)=2 πsinνa ν, (8.150) where Rn(ν)=a π(−1)nq2n 2n+11F1(2n+1;2n+2;iνa)+ a π(−1)nq2n 2n+11F1(2n+1;2n+2;−iνa).(8.151) Thesu mofth eKummerfunctionsm aybesimpliﬁe dto 1F1(2n+1;2n+2;iνa)+ 1F1(2n+1;2n+2;−iνa) =−i(−1)n(2n+1)!2n/summationdisplay k=0ik(νa)k−2n−1 k!/braceleftBig (−1)keiνa−e−iνa/bracerightBig .(8.152) Beforeturningtotheanalysisof(8.145),weexpandthatpartofthe integrandappearingin(8.145)inaserie sofLegendrepolynomials(see[14]) J0(νbsinθ)e±νbcosθ=∞/summationdisplay n=0(νb)n n!(−1)nPn(cosθ). (8.153) Bythemethodsde velope dinSection2.1,Equations(8.145)an d(8.146) may be transformed to ∞/summationdisplay n=0xncos/parenleftbigg n+1 2/parenrightbigg θ =/braceleftbigg cos1 2θ−/integraltext∞ 0G(ν)e−νbcosθsin/parenleftbig1 2θ−νbsinθ/parenrightbig dν, θ<θ 0 0, θ>θ 0(8. 154) ©200 1 CRC Press LLC wherewehaveusedtheseries(derivedthegeneratin gfunction ,seeAppendix , (B. 138)) ∞/summationdisplay n=0(−1)n n!(νb)ncos/parenleftbigg n+1 2/parenrightbigg θ=e−νbcosθsin/parenleftbigg1 2θ−νbsinθ/parenrightbigg .(8. 155) From (8. 154) we may derive the companion integro-series equation in alge- braic form xm+/integraldisplay∞ 0G(ν)Sm(ν)dν=Q0m(θ0), (8. 156) wherem= 0,1,2..., and Sm(ν) =2 π/integraldisplayθ0 0e−νbcosθsin/parenleftbigg1 2θ−νbsinθ/parenrightbigg cos/parenleftbigg m+1 2/parenrightbigg θ dθ. (8. 157) The structure of Equations (8. 150) and (8. 156) is interesting. If the contribution from the functions RmandSmare neglected, then the closed form solutions are precisely those previously obtained for the isolated disc and isolated spherical cap, respectively. The contribution from the functionsR mandSmmay be regarded as perturbation terms (though, as we shall see, not necessarily small in magnitude). The simultaneous solution of Equations (8. 150) and (8. 156) provides the potential of the coupled two-component structure. It is clear that a second-kind Fredholm equation for Gmay be obtained by elimination of the terms involvingx n; equally, a second-kind i.s.l.a.e. for the sequence {xn}∞ n=0may be obtained by elimination of the function G.Using (8. 150) to eliminate G, this i.s.l.a.e. is xm−∞/summationdisplay n=0x2nαnm(q,θ0) =Q0m(θ0)−βm(q,θ0), (8. 158) wherem= 0,1,2,..., and αnm(q,θ0) = 4 (−1)n π2/integraldisplayθ0 0cos/parenleftbigg m+1 2/parenrightbigg θ/braceleftbigg/integraldisplayq 0t2nt2sin3 2θ−sin1 2θ t4+ 2t2cos 2θ+ 1dt/bracerightbigg dθ, (8. 159) βm(q,θ0) =1 π2/integraldisplayθ0 0cos/parenleftbigg m+1 2/parenrightbigg θ× /braceleftbigg 2 arctan/bracketleftbigg2qcosθ 1−q2/bracketrightbigg sin1 2θ−cos1 2θln/bracketleftbigg1 + 2qcosθ+q2 1−2qcosθ+q2/bracketrightbigg/bracerightbigg dθ. (8. 160) In a similar way, we may deduce that Gsatisﬁes the second-kind integral equation G(µ)−/integraldisplay∞ 0G(ν)H(ν,µ;q,θ0) =2 πsinµa µ−L(µ;q,θ0), (8. 161) ©200 1 CRC Press LLC ©2001 CRC Press LLC where H(ν,µ;q,θ0) =4 π2b/integraldisplayq 0dtcosµbt× /integraldisplayθ0 0dθe−νbcosθsin/parenleftbigg1 2θ−νbsinθ/parenrightbiggt2cos3 2θ+ cos1 2θ t4+ 2t2cos 2θ+ 1,(8. 162) and L(µ;q,θ0) =−b π2/integraldisplayq 0dtcosµbt× /braceleftbigg θ0+ arctan/bracketleftbigg1−t2 1 +t2tanθ0/bracketrightbigg +1 2tln/bracketleftbigg1 + 2tsinθ0+t2 1−2tsinθ0+t2/bracketrightbigg/bracerightbigg .(8. 163) When the disc is much smaller than the radius of curvature of the cap (q/lessmuch1), it is possible to obtain an approximate analytical solution. In this limiting case the matrix elements can be factored as αnm=ξnβ∗ m(q,θ0)/parenleftbig 1 +O(q3)/parenrightbig , (8. 164) where ξn= (−1)nq2n 2n+ 1, β∗ m(q,θ0) =2 π2q/bracketleftbiggcos(m+ 1)θ0−1 m+ 1−cosmθ0−1 m/bracketrightbigg . (8. 165) It should be noted that βm=β∗ m(q,θ0) +O(q3).For the given approximation (q/lessmuch1,θ0arbitrary), the solution of the i.s.l.a.e. (8. 158) is xm= (C−1)β∗ m(q,θ0) +Q0m(θ0) +O(q3), (8. 166) whereC=/summationtext∞ n=0x2nξn.The value of Cis readily computed from (8. 166); the ﬁnal solution is xm=Q0m(θ0)−2 π2q1−Q00(θ0) 1 +4 π2qsin21 2θ0/bracketleftbiggcos(m+ 1)θ0−1 m+ 1−cosmθ0−1 m/bracketrightbigg . (8. 167) The total charge Qaccumulated on both components is the sum of that accumulated on the disc/parenleftbig Qd/parenrightbig and of that accumulated on the cap ( Qc) ; these are simply Qc=bx0, Qd=G(0). (8. 168) Using (8. 167) and (8. 150) at ν= 0,we deduce Qc=b/braceleftbigg Q00(θ0) +2 π2q1−cosθ0 1 +2 π2q(1−cosθ0)+O(q3)/bracerightbigg (8. 169) ©200 1 CRC Press LLC ©200 1 CRC Press LLC θ0(deg.)Qc/b Qd/aQ/b 0◦0 0.6366 0.0636 10◦0.1111(0.1108) 0.5661 0.1677 20◦0.2209(0.2200) 0.4963 0.2705 30◦0.3276(0.3258) 0.4285 0.3722 40◦0.4294(0.4268) 0.3636 0.4683 50◦0.5248(0.5216) 0.3028 0.5582 60◦0.6125(0.6090) 0.2467 0.6407 70◦0.6916(0.6880) 0.1961 0.7147 80◦0.7613(0.7579) 0.1514 0.7797 90◦0.8213(0.8183) 0.1131 0.8354 Table 8.1 Normalised value of total charge Q/b=(Qc+Qd)/b.The parameter q=a/b=0.1. and Qd=2 πa/braceleftbigg 1−Q00(θ0)−2 π2q(1−Q00(θ0))(1−cosθ0)/bracerightbigg +2 πaq2/braceleftbigg4 π2(1−cosθ0)2(1−Q00(θ0))+1 3Q02(θ0)/bracerightbigg +O(q3).(8.170) Someresult sofcalculatio nbasedontheseapproximat eformulaeforq=0.1 areshowninTable8.1.Theresult sforanisolate dspherica lcapareshown for comparison in brackets in the ﬁrst column of the table. The presence of the charged disc has a discernible eﬀect on the spherical cap even when it is small, increasing the charge on the cap. Rather more noticeable is the decrease in charge on the disc as the cap size increases; asthe angleθ 0increases, the disc is increasingly shielded by the larger charged conductor, and its surface charge distribution is correspondingly modiﬁed. More generally, whatever the values of the parameters qandθ0,the reg- ularised second-kind Equations (8. 158) and (8. 161) are readily solved bystandard numerical methods, and the behaviour of the coupled disc-cap struc-ture can be determined as a function of the parameters. If recursion formulaefor the coeﬃcients α nmandβmare exploited, a highly eﬃcient computational algorithm can be obtained for computation. ©200 1 CRC Press LLC Appendix A Notation TheKronecker symbol is deﬁned by δnm=/braceleftbigg1,n=m 0,n/negationslash=m. The order notation f(x) =O(g(x)) asx→a,means that |f(x)/g(x)| remains bounded as x→a.(This includes the possibilities a=±∞.) Simi- larly, the notation an=O(bn) asn→ ∞ means |an/bn|remains bounded as n→ ∞. TheHeaviside function is deﬁned by H(x) =/braceleftbigg1,x> 0 0,x< 0. ©200 1 CRC Press LLC Appendix B Special Functions Only the most important relations for the special functions employed in this book are included in this section. For more detailed information, the readeris referred to standard works on the special functions including, for example,[59, 1, 57, 58], and a summary treatment in [27]. B.1 The Gamma function The Gamma function Γ deﬁned by Γ(z) =/integraldisplay∞ 0tz−1e−tdt, Re(z)>0 (B. 1) is a generalization of the factorial: when nis a nonnegative integer Γ(n+ 1) =n! Therecurrence formula for the factorial is Γ(z+ 1) =zΓ(z), (B. 2) and the reﬂection formula is Γ(z)Γ(1−z) =π sin(πz), (B. 3) from which it follows that Γ(1 2) =√π; the duplication formula is Γ(2z) = (2π)−1 222z−1 2Γ(z)Γ(z+1 2). (B. 4) Two asymptotic formulae are widely used. Stirling’s formula states Γ(z)/revsimilare−zzz−1 2(2π)1 2/bracketleftbigg 1 +1 12z+1 288z2−139 51840z3−571 2488320z4+.../bracketrightbigg , (B. 5) ©200 1 CRC Press LLC whenz→ ∞ in|argz|< π; Field’s formula states that the ratio of Gamma functions has an asymptotic expansion of the form for suitable cn, Γ(z+a) Γ(z+b)/revsimilarza−b∞/summationdisplay n=0cnΓ(b−a+n) Γ(b−a)1 zn, (B. 6) whenz→ ∞ andz/negationslash=−a,−a−1,...;z/negationslash=−b,−b−1,.... The ﬁrst few terms in the expansion are Γ(z+a) Γ(z+b)=za−b(1 +(a−b)(a+b−1) 2z+ 1 12/parenleftbigga−b 2/parenrightbigg (3(a+b−1)2−a+b−1)1 z2+...).(B. 7) Closely connected with the Gamma function is the Beta function deﬁned for Re(p)>0,Re(q)>0; it equals B(p,q) =/integraldisplay1 0tp−1(1−t)q−1dt=Γ(p)Γ(q) Γ(p+q). (B. 8) B.2 Hypergeometric functions The generalised hypergeometric function is deﬁned by pFq(a1,...,a p;b1,...,b q;z)≡∞/summationdisplay k=0(a1)k(a2)k....(ap)k (b1)k(b2)k...(bq)k·zk k!(B. 9) where the notation for the Pochhammer symbol (a)kdef=a(a+ 1)...(a+k−1) ; (a)0def= 1 (B. 10) has been used; the upper parameters− →a= (a1,...,a p) are unrestricted, whereas the lower parameters− →b= (b1,...,b q) are restricted so that bj/negationslash= 0,−1,−2,.... Note that when a/negationslash= 0,−1,−2,..., (a)k=Γ (a+k) Γ (a). (B. 11) Whenp≤q, the series converges for all complex z.Whenp=q+ 1,the series has radius of convergence 1, converging inside the unit disc |z|<1; it converges on the unit disc |z|= 1 provided Re q/summationdisplay k=1bk−q+1/summationdisplay j=1aj>0, (B. 12) ©200 1 CRC Press LLC or alternatively, it converges everywhere on the unit disc, except at the point z= 1,provided −1<Re q/summationdisplay k=1bk−q+1/summationdisplay j=1aj≤0. (B. 13) If the one of upper parameters is equal to zero or a negative integer, then the series terminates and is a hypergeometric polynomial. The function 1F1(a;b;z)≡M(a,b,z ) is known as Kummer’s function; many special functions are expressible as Kummer’s function with particular param-eters [1, 59]. The Gaussian hypergeometric series is a special case of the hypergeometric function with p= 2,q= 1, 2F1(a,b;c;z) =∞/summationdisplay k=0(a)k(b)k (c)k·zk k!. (B. 14) It satisﬁes the diﬀerential equation z(1−z)d2U dz2+ [c−(a+b+ 1)z]dU dz−abU= 0. (B. 15) Whenaorbis equal to a negative integer, then the series (B. 14) terminates and is a hypergeometric polynomial; ifa=−m(ma positive integer), F(−m,b;c;z) =m/summationdisplay n=0(−m)n(b)n (c)nzn n!. (B. 16) This formula is also well deﬁned when c=−m−l, l= 0,1,2,... F(−m,b;−m−l;z) =m/summationdisplay n=0(−m)n(b)n (−m−l)nzn n!. (B. 17) Many special functions are particular examples of the Gaussian hypergeo- metric series (B. 14) with appropriate arguments, including the Jacobi poly- nomials discussed in the next section. Hypergeometric functions satisfy agreat number of transformation rules (see [1]) that provide many interestingand useful connections between the various special functions. B.3 Orthogonal polynomials: Jacobi polynomials, Leg- endre polynomials Jacobi polynomials and Legendre polynomials are two families of classical orthogonal polynomials whose properties are extensively described in [58]. ©200 1 CRC Press LLC ©200 1 CRC Press LLC For each ﬁxed ( α,β) withα >−1,β >−1,theJacobi polynomials P(α,β) n are polynomials of degree n(= 0,1,2,...), and are orthogonal with respect to theweighted scalar product on [−1,1] employing the weight function wα,β(x) = (1−x)α(1 +x)β: (P(α,β) n,P(α,β) m) =/integraldisplay1 −1(1−x)α(1 +x)βP(α,β) n(x)P(α,β) m(x)dx=h(α,β) nδnm. (B. 18) The polynomials are normalised by their value at x= 1, P(α,β) n(1) =/parenleftbiggn+α n/parenrightbigg =Γ(n+α+ 1) Γ(n+ 1)Γ(α+ 1), (B. 19) so that their squared norm is h(α,β) n =/vextenddouble/vextenddouble/vextenddoubleP(α,β) n/vextenddouble/vextenddouble/vextenddouble2 =2α+β+1 2n+α+β+ 1Γ(n+α+ 1)Γ(n+β+ 1) n!Γ(n+α+β+ 1).(B. 20) Jacobi polynomials may also be normalised by the requirement that the weighted scalar product be equal to unity when n=m; the members of this orthonormal family are denoted ˆP(α,β) n =/braceleftBig h(α,β) n/bracerightBig−1 2P(α,β) n. In common with all the families of classical orthogonal polynomials, the Jacobi polynomials satisfy a recurrence relation of form pn+1−(anx+bn)pn+cnpn−1= 0, n = 1,2,... (B. 21) For the Jacobi polynomials pn=P(α,β) n, the coeﬃcients an,bn,cnand the two lowest degree polynomials are an=(2n+α+β+ 1)(2n+α+β+ 2) (2n+ 2)(n+α+β+ 1), bn=(2n+α+β+ 1)(α2−β2) (2n+ 2)(n+α+β+ 1)(2n+α+β), (B. 22) cn=2(n+α)(n+β)(2n+α+β+ 2) (2n+ 2)(n+α+β+ 1)(2n+α+β), and P(α,β) 0(x) = 1,P(α,β) 1(x) =1 2(α−β) +/bracketleftbigg 1 +1 2(α+β)/bracketrightbigg x. (B. 23) They satisfy the diﬀerential equation (1−x2)d2y dx2+ [β−α−(α+β+ 2)x]dy dx+n(n+α+β+ 1)y= 0.(B. 24) ©200 1 CRC Press LLC ©200 1 CRC Press LLC Comparing this with the diﬀerential Equation (B. 15) for the Gaussian hy- pergeometric series and making the transformation z=1 2(1−x) leads us to make the identiﬁcation a=−n, b=n+α+β+ 1, c=α+ 1 and to recognise that P(α,β) n(x) is the hypergeometric polynomial P(α,β) n(x) =/parenleftbiggn+α n/parenrightbigg 2F1/parenleftbigg −n,n+α+β+ 1;α+ 1;1−x 2/parenrightbigg . (B. 25) Thus, an explicit form for P(α,β) n(x) is Γ(n+α+ 1) n!Γ(α+ 1)n/summationdisplay m=0(−n)m(n+α+β+ 1) m m!(α+ 1) m/parenleftbigg1−x 2/parenrightbiggm =Γ(n+α+ 1) n!Γ(n+α+β+ 1)n/summationdisplay k=0Γ(n+ 1)Γ(n+k+α+β+ 1) Γ(k+ 1)Γ(n−k+ 1)Γ(k+α+ 1)/parenleftbiggx−1 2/parenrightbiggk . From the symmetry property P(α,β) n(−x) = (−1)nP(β,α) n(x), (B. 26) one obtains the alternative representation P(α,β) n(x) = (−1)n/parenleftbiggn+β n/parenrightbigg 2F1/parenleftbigg −n,n+α+β+ 1;β+ 1;1 +x 2/parenrightbigg .(B. 27) Many other representations are possible because of the great number of trans- formation relations that the hypergeometric function satisﬁes. The Jacobi polynomials satisfy Rodrigues’ formula P(α,β) n(x) =(−1)n 2nn!1 (1−x)α(1 +x)β/parenleftbiggd dx/parenrightbiggn/bracketleftbig (1−x)α+n(1 +x)β+n/bracketrightbig , (B. 28) from which follows the useful relation −2n(1−x)α(1 +x)βP(α,β) n(x) =d dx/bracketleftBig (1−x)α+1(1 +x)β+1P(α+1,β+1) n−1 (x)/bracketrightBig . (B. 29) Thediﬀerential relation expresses derivatives in terms of polynomials of the same parameters ( α,β) : (2n+α+β)(1−x2)d dxP(α,β) n(x) =n[α−β−(2n+α+β)x]P(α,β) n(x) + 2(n+α)(n+β)P(α,β) n−1(x).(B. 30) ©200 1 CRC Press LLC Other recurrence relations connect polynomials with indices ( α,β) to those with indices ( α+ 1,β) and (α,β+ 1), /parenleftbigg n+α 2+β 2+ 1/parenrightbigg (1−x)P(α+1,β) n (x) = (n+α+ 1)P(α,β) n(x)−(n+ 1)P(α,β) n+1(x),(B. 31) /parenleftbigg n+α 2+β 2+ 1/parenrightbigg (1 +x)P(α,β+1) n (x) = (n+β+ 1)P(α,β) n(x) + (n+ 1)P(α,β) n+1(x),(B. 32) 2P(α,β) n(x) = (1 −x)P(α+1,β) n (x) + (1 +x)P(α,β+1) n (x); (B. 33) alsorecurrence relations between polynomials with indices ( α,β) and those with indices ( α−1,β) and (α,β−1) (2n+α+β)P(α−1,β) n (x) = (n+α+β)P(α,β) n(x)−(n+β)P(α,β) n−1(x),(B. 34) (2n+α+β)P(α,β−1) n (x) = (n+α+β)P(α,β) n(x) + (n+α)P(α,β) n−1(x),(B. 35) P(α,β−1) n (x)−P(α−1,β) n (x) =P(α,β) n−1(x). (B. 36) These relations may be used to extend the deﬁnition of Jacobi polynomials for parameters ( α,β) whereα≤ −1 orβ≤1; in the text, the most commonly encountered examples are P(−1,0) n (x) =1 2(Pn(x)−Pn−1(x)), (B. 37) P(0,−1) n =1 2(Pn(x) +Pn−1(x)). (B. 38) The generating function is F(z,x) =∞/summationdisplay n=0P(α,β) n(x)zn= 2α+βR−1(1−z+R)−α(1 +z+R)−β,(B. 39) whereR=√ 1−2xz+z2,the branch being ﬁxed by specifying R= 1 when z= 0; the power series is convergent when |z|<1. For particular values of α,βthere are other generating functions. An asymptotic formula withα,β,x ﬁxed andn→ ∞ is P(α,β) n(cosθ) =cos/parenleftbig/bracketleftbig n+1 2(α+β+ 1)/bracketrightbig θ−π 4(2α+ 1)/parenrightbig √πn/parenleftbig sin1 2θ/parenrightbigα+1 2/parenleftbig cos1 2θ/parenrightbigβ+1 2+O/parenleftBig n−3 2/parenrightBig (B. 40) where 0<θ<π . ©200 1 CRC Press LLC Manyofth eclassicalorthogonalpolynomialsareparticularexample sof Jacobipolynomials,includingtheLegendrepolynomials Pn=P(0,0) n,the Chebyshevpolynomialsofﬁrs tkind Tn=Γ/parenleftbig1 2/parenrightbig Γ(n+1) Γ/parenleftbig n+1 2/parenrightbigP(−1 2,−1 2) n, (B.41) theChebyshevpolynomial sofsecondkind Un=Γ/parenleftbig3 2/parenrightbig Γ(n+1) Γ/parenleftbig n+3 2/parenrightbigP(1 2,1 2) n, (B.42) andtheGegenbauerpolynomials Cγ n=(2γ)n/parenleftbig γ+1 2/parenrightbig nP(γ−1 2,γ−1 2) n. (B.43) Thusifnisanonnegati veinteger, cosnθ=Γ/parenleftbig1 2/parenrightbig Γ(n+1) Γ/parenleftbig n+1 2/parenrightbigP(−1 2,−1 2) n (cosθ), (B.44) sinnθ=Γ/parenleftbig3 2/parenrightbig Γ(n+1) Γ/parenleftbig n+1 2/parenrightbigsinθP(1 2,1 2) n−1(cosθ). (B.45) Explicitformsforothertrigonometricfunction sare cos(n+1 2)θ=Γ/parenleftbig1 2/parenrightbig Γ(n+1) Γ/parenleftbig n+1 2/parenrightbigcos1 2θP(−1 2,1 2) n (cosθ), (B.46) sin(n+1 2)θ=Γ/parenleftbig1 2/parenrightbig Γ(n+1) Γ/parenleftbig n+1 2/parenrightbigsin1 2θP(1 2,−1 2) n (cosθ). (B.47) B.3.1TheassociatedLegendr epolynomials. Whenn/greaterorequalslantm,therelationshipbe tweentheassociatedLegendr efunction sPm n andtheJacob ipolynomials P(m,m ) n−mis Pm n(cosθ)=2−msinmθΓ(n+m+1) Γ(n+1)P(m,m ) n−m(cosθ)(B .48) andtheconnectio nwithLegendrepolynomialsis Pm n(x)=/parenleftbig 1−x2/parenrightbigm 2dm dxmPn(x). (B.49) AnotherorthonormalfamilyofJacobipolynomials( n≥k,kﬁxed),considered inChapter2hasth eform ˆP(k−1 2,k+1 2) n−k(cosθ) =(−1)k √π/braceleftbigg(n−k)! (n+k)!/bracerightbigg1 2/parenleftbigg1 sinθd dθ/parenrightbiggk/bracketleftBigg cos/parenleftbig n+1 2/parenrightbig θ cos1 2θ/bracketrightBigg . (B. 50) ©200 1 CRC Press LLC B.3.2 The Legendre polynomials. The Legendre polynomials Pn(x) form a subclass of the associated Legendre functionsPm ν(z) (wherem= 0 ,ν=n= 0,1,2,...andz=xis real, −1≤x≤1 ) that are considered in the next subsection and so all properties of these functions are valid for the Legendre polynomials. In the context of classical orthogonal polynomials, the Legendre polynomials are the Jacobipolynomials with α=β= 0. Thus, they are orthogonal with respect to the constant (unit) function, are normalised by the condition P n(1) = 1,and have square norm where hn=h(0,0) n=/bardblPn/bardbl2= 2 (2n+ 1)−1. They satisfy therecurrence relation (n+ 1)Pn+1(x)−(2n+ 1)xPn(x) +nPn−1(x) = 0, n = 1,2,... (B. 51) whereP0(x) = 1, P1(x) =x.ThusP2(x) =3 2x2−1 2.They satisfy the diﬀer- ential equation (1−x2)d2y dx2−2xdy dx+n(n+ 1)y= 0, (B. 52) and have the hypergeometric polynomial representation Pn(x) = 2F1/parenleftbigg −n,n+ 1; 1;1−x 2/parenrightbigg . (B. 53) The Rodrigues’ formula is simply Pn(x) =1 2nn!/parenleftbiggd dx/parenrightbiggn/bracketleftbig (x2−1)n/bracketrightbig . (B. 54) Useful diﬀerential and integration relations are (1−x2)d dxPn(x) =n[Pn−1(x)−xPn(x)] = (n+ 1) [xPn(x)−Pn+1(x)] (B. 55) nPn(x) =xd dxPn(x)−d dxPn−1(x) (B. 56) (n+ 1)Pn(x) =d dxPn+1(x)−xd dxPn(x) (B. 57) (2n+ 1)/integraldisplay Pn(x)dx=Pn+1(x)−Pn−1(x). (B. 58) Two generating functions are ∞/summationdisplay n=0Pn(x)zn= (1−2xz+z2)−1,−1<x< 1,|z|<1, (B. 59) ∞/summationdisplay n=01 n!Pn(cosθ)zn=ezcosθJ0(zsinθ) (B. 60) ©200 1 CRC Press LLC The asymptotic formula for the Legendre polynomials when n→ ∞ is Pn(cosθ) =Γ(n+ 1) Γ(n+3 2)(1 2πsinθ)−1 2cos/bracketleftbigg/parenleftbigg n+1 2/parenrightbigg θ−π 4/bracketrightbigg +O/parenleftbig n−1/parenrightbig (B. 61) wherex= cosθis ﬁxed and 0 <θ<π . B.4 Associated Legendre functions Associated Legendre functions of degree νand orderµare solutions of complex argument zof the diﬀerential equation (1−z2)d2y dz2−2zdy dz+/bracketleftbigg ν(ν+ 1)−µ2 1−z2/bracketrightbigg y= 0. (B. 62) The constants νandµare in general arbitrary complex numbers. The singu- larities of the diﬀerential equation are located at z=±1,∞and are regular. We shall consider ﬁrst the ordinary Legendre functions of degree νcorrespond- ing to the choice µ= 0,and subsequently consider the associated Legendre functions of nonzero order µ,restricting it to be integral. B.4.1 Ordinary Legendre functions Whenµ= 0, the diﬀerential equation becomes (1−z2)d2y dz2−2zdy dz+ν(ν+ 1)y= 0. (B. 63) A pair of linearly independent solutions is the ﬁrst-kind and second-kind Leg- endre functions denoted Pν(z) andQν(z); they are entire functions of zin the plane cut along ( −∞,1]. The ﬁrst-kind function is deﬁned by Pν(z) = 2F1/parenleftbigg −ν,ν+ 1; 1;1−z 2/parenrightbigg ,|arg(z+ 1)|<π. (B. 64) It possesses the symmetry property P−ν−1=Pν. An alternative representa- tion forPνthat is useful for large zis Pν(z) =(2z)−ν−1Γ(−1 2−ν)√πΓ(−ν)2F1/parenleftbiggν 2+ 1,ν+ 1 2;ν+3 2;1 z2/parenrightbigg + (2z)νΓ(ν+1 2) Γ(ν+ 1)2F1/parenleftbigg1−ν 2,−ν 2;1 2−ν;1 z2/parenrightbigg ,(B. 65) ©200 1 CRC Press LLC validwhen |z|>1,|argz|<π,ν /negationslash=±1 2,±3 2,....Anotherusefulreprese ntation is Pν(z)=Γ/parenleftbigν 2+1 2/parenrightbig √πΓ/parenleftbigν 2+1/parenrightbigcosνπ 22F1/parenleftbigg1+ν 2,−ν 2;1 2−ν;z2/parenrightbigg + 2Γ/parenleftbigν 2+1/parenrightbig √πΓ/parenleftbigν 2+1 2/parenrightbigsinνπ 2z2F1/parenleftbigg1−ν 2,ν 2+1;3 2;z2/parenrightbigg ,(B.66) validwhe n|z|<1,andνisarbitrary. Thesecond-kin dLegendrefunctionisdeﬁne dby Qν(z)=√πΓ(ν+1) Γ/parenleftbig ν+3 2/parenrightbig (2z)ν+12F1/parenleftbiggν 2+1,ν 2+1 2;ν+3 2;z−2/parenrightbigg , (B.67) whereν/negationslash=−1,−2,...;itpossessesananalyticconti nuationinth eentire complexplane,excludin gthepointsz=±1,withabran chcutalong( −∞,1]. Anotherusefulexpansionis Qν(z)=e∓iνπ/ 2√πΓ/parenleftbigν 2+1/parenrightbig Γ/parenleftbigν 2+1 2/parenrightbigz2F1/parenleftbigg1−ν 2,1+ν 2;3 2;z2/parenrightbigg ∓ e∓iνπ/ 2i√πΓ/parenleftbigν 2+1 2/parenrightbig 2Γ/parenleftbigν 2+1/parenrightbig2F1/parenleftbigg1+ν 2,−ν 2;1 2;z2/parenrightbigg ,(B.68) validwhen |z|<1,ν/negationslash=−1,−2,...,theup persignbein gtakenwhenIm z>0, andthel owersignwhe nImz<0. TheWronskian is W{Pν(z),Qν(z)}=P/prime ν(z)Qν(z)−Pν(z)Q/prime ν(z)=/parenleftbig 1−z2/parenrightbig−1. (B.69) Thefollowingformulaeareparticularlyusefulforestimationofth easymp- toticallysmallparametersencountere dinChapter s3an d4. Qν(coshα) =√πΓ(ν+ 1) Γ(ν+3 2)e−(ν+1)αF/parenleftbigg ν+ 1,1 2;ν+3 2;e−2α/parenrightbigg (B. 70) Pν(coshα) =Γ(ν+ 1)√πΓ(ν+3 2)tan (νπ)e−(ν+1)αF/parenleftbigg ν+ 1,1 2;ν+3 2;e−2α/parenrightbigg +Γ(ν+1 2)√πΓ(ν+ 1)eναF/parenleftbigg −ν,1 2;1 2−ν;e−2α/parenrightbigg ,(B. 71) whereν/negationslash=±1 2,±3 2,.... Asymptotic expansions valid when |ν| → ∞,|argν| ≤π 2−δ,andαis ﬁxed (0<α< ∞) are Pν(coshα) =e(ν+1 2)α √ 2νπsinhα/bracketleftBig 1 +O/parenleftBig |ν|−1/parenrightBig/bracketrightBig (B. 72) ©200 1 CRC Press LLC Qν(coshα) =√π√ 2νsinhαe−(ν+1 2)α/bracketleftBig 1 +O/parenleftBig |ν|−1/parenrightBig/bracketrightBig ; (B. 73) whenνis real and ν→ ∞,andθis ﬁxed in the interval δ≤θ≤π−δ(for someδ>0), Pν(cosθ) =/radicalbigg 2 νπsinθsin/bracketleftbigg (ν+1 2)θ+1 4π/bracketrightbigg/bracketleftBig 1 +O/parenleftBig |ν|−1/parenrightBig/bracketrightBig , (B. 74) Qν(cosθ) =/radicalbigg 2 νπsinθcos/bracketleftbigg (ν+1 2)θ+1 4π/bracketrightbigg/bracketleftBig 1 +O/parenleftBig |ν|−1/parenrightBig/bracketrightBig . (B. 75) Explicit expressions are P0(z) = 1, Q 0(z) =1 2ln/parenleftbiggz+ 1 z−1/parenrightbigg , (B. 76) P1(z) =z, Q 1(z) =z 2ln/parenleftbiggz+ 1 z−1/parenrightbigg −1; (B. 77) these are valid when ztakes real values x∈(−1,1). P±1 2andQ±1 2are closely related to complete elliptic integrals of the ﬁrst kind K(k) =/integraldisplayπ 2 0dθ/radicalbig 1−k2sin2θ(B. 78) and of the second kind E(k) =/integraldisplayπ 2 0/radicalbig 1−k2sin2θdθ, (B. 79) the properties of which are discussed in [59, 1]; in particular [14] P−1 2(z) =2 π/radicalbigg 2 z+ 1K/parenleftBigg/radicalbigg z−1 z+ 1/parenrightBigg , (B. 80) Q−1 2(z) =/radicalbigg 2 z+ 1K/parenleftBigg/radicalbigg 2 z+ 1/parenrightBigg , (B. 81) P1 2(z) =2 π/parenleftBig z+/radicalbig z2−1/parenrightBig1 2E/parenleftBigg/radicalBigg 2(z2−1)1/2 z+ (z2−1)1/2/parenrightBigg , (B. 82) Q1 2(z) =z/radicalbigg 2 z+ 1K/parenleftBigg/radicalbigg 2 z+ 1/parenrightBigg −/radicalbig 2(z+ 1)E/parenleftBigg/radicalbigg 2 z+ 1/parenrightBigg . (B. 83) ©200 1 CRC Press LLC Whenz=xis real and −1<x< 1,these become P−1 2(x) =2 πK/parenleftBigg/radicalbigg 1−x 2/parenrightBigg , (B. 84) Q−1 2(x) =K/parenleftBigg/radicalbigg 1 +x 2/parenrightBigg , (B. 85) P1 2(x) =2 π/bracketleftBigg 2E/parenleftBigg/radicalbigg 1−x 2/parenrightBigg −K/parenleftBigg/radicalbigg 1−x 2/parenrightBigg/bracketrightBigg , (B. 86) Q1 2(x) =K/parenleftBigg/radicalbigg 1 +x 2/parenrightBigg −2E/parenleftBigg/radicalbigg 1 +x 2/parenrightBigg . (B. 87) Whenz= coshαis real and exceeds 1 ,these become P−1 2(coshα) =/parenleftBigπ 2coshα 2/parenrightBig−1 K/parenleftBig tanhα 2/parenrightBig , (B. 88) Q−1 2(coshα) = 2e−α/2K/parenleftbig e−α/parenrightbig , (B. 89) P1 2(coshα) =2 πeα/2E/parenleftBig/radicalbig 1−e−2α/parenrightBig , (B. 90) Q1 2(coshα) =/parenleftBig 2 coshα 2−sechα 2/parenrightBig K/parenleftBig sechα 2/parenrightBig −2 coshα 2E/parenleftBig sechα 2/parenrightBig , (B. 91) Another useful result is Q−1 2(cosh 2σ) = sechσK(sechσ). (B. 92) Integral representations valid for any complex νand Re cosh α>0 are Pν(coshα) =/integraldisplayα 0cosh/parenleftbig ν+1 2/parenrightbig θ√ 2 coshα−2 coshθdθ, (B. 93) and the Mehler-Dirichlet formula ([55]) Pν(cosβ) =2 π/integraldisplayβ 0cos/parenleftbig ν+1 2/parenrightbig θ√2 cosθ−2 cosβdθ. (B. 94) Whenα>0,and−1<Reν <1, Pν(coshα) =2 πcot/parenleftbigg ν+1 2/parenrightbigg π/integraldisplay∞ αsinh/parenleftbig ν+1 2/parenrightbig θ√ 2 coshθ−2 coshαdθ. (B. 95) Also, when Re ν >−1, Qν(coshα) =/integraldisplay∞ αe−(ν+1 2)θ √ 2 coshθ−2 coshαdθ. (B. 96) ©200 1 CRC Press LLC A deﬁnite integral that frequently occurs is /integraldisplayz0 −1Q−1 2(z)Pm(z)dz= 1−z2 0/parenleftbig m+1 2/parenrightbig2/braceleftBig Pm(z0)Q/prime −1 2(z0)−P/prime m(z0)Q−1 2(z0)/bracerightBig .(B. 97) (It is evaluated using integration by parts and the deﬁning diﬀerential equa- tions for these functions.) B.4.2 Conical functions The Legendre functions P−1 2+iτandQ−1 2+iτwith realτoccur in boundary value problems in conical geometry. The function P−1 2+iτ(cosφ) =P−1 2−+iτ(cosφ) is real for real φ,as may be seen from its hypergeometric representation de- rived from (B. 64), P−1 2+iτ(cosφ) = 2F1/parenleftbigg1 2+iτ,1 2−iτ; 1; sin21 2φ/parenrightbigg (B. 98) AlthoughP−1 2+iτandQ−1 2+iτare linearly independent solutions of the dif- ferential equation, the functions P−1 2+iτ(x) andP−1 2+iτ(−x) are also linearly independent. The Wronskians are W/parenleftBig P−1 2+iτ(x),P−1 2+iτ(−x)/parenrightBig =P−1 2+iτ(x)P/prime −1 2+iτ(−x)−P−1 2+iτ(−x)P/prime −1 2+iτ(x) =2 πcosh (πτ)W/parenleftBig P−1 2+iτ(x),Q−1 2+iτ(x)/parenrightBig =2 πcosh (πτ)/parenleftbig 1−x2/parenrightbig−1.(B. 99) P−1 2+iτhas the integral representation P−1 2+iτ(coshx) =√ 2 π/integraldisplayx 0cosτt dt√ coshx−cosht, =√ 2 πcoth(πτ)/integraldisplay∞ xsinτt dt√ cosht−coshx.(B. 100) When asτ→ ∞, P−1 2+iτ(cosθ)/revsimilareτθ √ 2πτsinθ, (B. 101) uniformly in the sector δ≤θ≤π−δ. ©200 1 CRC Press LLC B.4.3 Associated Legendre functions of integer order The conventional choice for a pair of linearly independent solutions to the diﬀerential Equation (B. 62) employs the ﬁrst-kind and second-kind associated Legendre functions denoted Pµ ν(z) andQµ ν(z) and deﬁned by Pµ ν(z) =1 Γ (1−µ)/parenleftbiggz+ 1 z−1/parenrightbigg1 2µ 2F1/parenleftbigg −ν,ν+ 1; 1−µ;1−z 2/parenrightbigg ,(B. 102) and Qµ ν(z) =√πeµπiΓ (ν+µ+ 1)/parenleftbig z2−1/parenrightbig1 2µ 2ν+1zν+µ+1Γ/parenleftbig ν+3 2/parenrightbig × 2F1/parenleftbigg1 2ν+1 2µ+ 1,1 2ν+1 2µ+1 2;ν+3 2;z−2/parenrightbigg .(B. 103) This is valid for the complex plane with a branch cut along ( −∞,1].When µis a positive integer, the Gamma function factor creates some diﬃculty; in this case the deﬁnitions of the associated Legendre functions of degree m(= 1,2,...) are taken to be Pm ν(z) =/parenleftbig z2−1/parenrightbig1 2mdm dzmPν(z) (B. 104) and Qm ν(z) =/parenleftbig z2−1/parenrightbig1 2mdm dzmQν(z). (B. 105) Whenz=x∈(−1,1) is real, it is convenient to modify these deﬁnitions in the fashion described in [27]. Pm ν, Qm νare generalisations of the Legendre polynomials Pn, Qn,reducing to them when m= 0 andν=n= 0,1,2,.... Pm ν(z) is an entire function of ν, whileQmν(z) is a meromorphic function ofνwith poles at the points ν=−1,−2,.... They have the hypergeometric function representations Pm ν(z) =Γ(ν+m+ 1) 2mΓ(m+ 1)Γ(ν−m+ 1)(z2−1)1 2m× 2F1/parenleftbigg m−ν,ν+ 1 +m;m+ 1;1−z 2/parenrightbigg ,(B. 106) valid when |z−1|<2,|arg(z−1)|<π, andνis arbitrary, and Qm ν(z) =(−1)m√πΓ(ν+m+ 1)(z2−1)1 2m 2ν+1zν+m+1Γ(ν+3 2)× 2F1/parenleftbiggν+m 2+ 1,ν+m+ 1 2;ν+3 2;1 z2/parenrightbigg ,(B. 107) ©200 1 CRC Press LLC valid when |z|>;|arg(z±1)|< π, andν/negationslash=−1,−2,.... Whenx∈(−1,1) is real, Pm ν(x) =(−1)mΓ(ν+m+ 1) 2mΓ(m+ 1)Γ(ν−m+ 1)(1−x2)1 2m× 2F1/parenleftbigg m−ν,ν+m+ 1;m+ 1;1−x 2/parenrightbigg .(B. 108) Both functions Pµ ν(z) andQµ ν(z) satisfy the same recurrence relations: Pµ+1 ν(z) = (z2−1)−1 2/bracketleftbig (ν−µ)zPµ ν(z)−(ν+µ)Pµ ν−1(z)/bracketrightbig , (B. 109) (ν−µ+ 1)Pµ ν+1(z) = (2ν+ 1)zPµ ν(z)−(ν+µ)Pµ ν−1(z),(B. 110) (z2−1)dPµ ν(z) dz=νzPµ ν(z)−(ν+µ)Pµ ν−1(z). (B. 111) Transformation formulae relate negative and positive indices: Pµ −ν−1(z) =Pµ ν(z), Pµ −ν−1(x) =Pµ ν(x),−1<x< 1; (B. 112) P−m ν(z) =Γ(ν−m+ 1) Γ(ν+m+ 1)Pm ν(z), P−m ν(x) = (−1)mΓ(ν−m+ 1) Γ(ν+m+ 1)Pm ν(x),−1<x< 1; (B. 113) Qµ −ν−1(z) =1 sinπ(ν−µ)/bracketleftbig −πeµπicosνπPµ ν(z) + sinπ(ν+µ)Qµ ν(z)/bracketrightbig ; (B. 114) Q−µ ν(z) =e−2µπiΓ(ν−µ+ 1) Γ(ν+µ+ 1)Qµ ν(z), (B. 115) Q−m ν(x) = (−1)mΓ(ν−m+ 1) Γ(ν+m+ 1)Qm ν(x). (B. 116) The Formulae (B. 114)–(B. 116) require that −1<x< 1 andν/negationslash=m−1,m− 2,.... Finally we note that when m>n, Pm n(z) =Pm n(x) = 0. (B. 117) Also Pm n(−x) = (−1)m+nPm n(x),−1<x< 1. (B. 118) TheWronskian is W{Pm ν(z),Qm ν(z)}=(−1)m (1−z2)Γ (ν+m+ 1) Γ (ν−m+ 1), (B. 119) ©200 1 CRC Press LLC W{Pm ν(x),Qm ν(x)}=1 (1−x2)Γ (ν+m+ 1) Γ (ν−m+ 1),−1<x< 1.(B. 120) Some explicit expressions are P−1 0(z) =P−1 −1(z) =/radicalbigg z−1 z+ 1, (B. 121) P1 1(x) =−/radicalbig 1−x2, P1 2(x) =−3x/radicalbig 1−x2. (B. 122) For ﬁxedz /∈(−∞,−1)∪(1,∞) and ﬁxed µ,as Re(ν)→ ∞ Pµ ν(z) =1√ 2π(z2−1)1/4Γ(ν+µ+ 1) Γ(ν+3 2)/bracketleftBig z+/radicalbig z2−1/bracketrightBigν+1 2× 2F1/parenleftBigg 1 2+µ,1 2−µ;3 2+ν;z+√ z2−1 2√ z2−1/parenrightBigg + 1√ 2π(z2−1)1/4Γ(ν+µ+ 1) Γ(ν+3 2)ie−iµπ/bracketleftBig z−/radicalbig z2−1/bracketrightBigν+1 2× 2F1/parenleftBigg 1 2+µ,1 2−µ;3 2+ν;−z+√ z2−1 2√ z2−1/parenrightBigg ,(B. 123) and for ﬁxed z /∈(−∞,−1) and ﬁxed µ,as Re(ν)→ ∞ Qµ ν(z) =eiµπ/radicalbiggπ 21 (z2−1)1/4Γ(ν+µ+ 1) Γ(ν+3 2)/bracketleftBig z−/radicalbig z2−1/bracketrightBigν+1 2× 2F1/parenleftBigg 1 2+µ,1 2−µ;3 2+ν;−z+√ z2−1 2√ z2−1/parenrightBigg .(B. 124) B.5 Bessel functions The commonly employed solutions of Bessel’s diﬀerential equation z2d2w dz2+zdw dz+ (z2−ν2)w= 0 (B. 125) are the Bessel functions of the ﬁrst kind Jν(z), of the second kind Yν(z) (also called the Neumann function), and of the third kind H(1) ν(z),H(2) ν(z) (also called the Hankel functions of the ﬁrst and second kind, respectively), deﬁned below;ν,zare in general complex. The classic treatise is Watson [73]. Each is a regular (holomorphic) function of zin the entire z- plane cut along the negative real axis; for ﬁxed z(/negationslash= 0) each is an entire function of ν. Whenνis integral,Jν(z) has no branch point and is an entire function of z. ©200 1 CRC Press LLC The series representation forJνis Jν(z) =1 Γ(ν+ 1)/parenleftBigz 2/parenrightBigν 0F1/parenleftbigg ν+ 1;−z2 4/parenrightbigg =∞/summationdisplay k=0(−1)k(z/2)2k+ν k!Γ(k+ν+ 1).(B. 126) Whenν=−nis a negative integer, and J−n(z) = (−1)nJn(z),for allz.The Neumann function is deﬁned by Yν(z) =1 sin (νπ)[Jν(z) cos (νπ)−J−ν(z)] (B. 127) where the right-hand side of this equation is replaced by its limiting value if νis an integer or zero. When ν=nis a nonnegative integer, Yn(z) =2 πJn(z) ln/parenleftBigz 2/parenrightBig −1 π/parenleftBigz 2/parenrightBig−nn−1/summationdisplay k=0(n−k−1)! k!/parenleftBigz 2/parenrightBig2k − 1 π/parenleftBigz 2/parenrightBign∞/summationdisplay k=0[ψ(k+ 1) +ψ(n+k+ 1)](−1)k k!(n+k)!/parenleftBigz 2/parenrightBig2k (B. 128) whereψ(k) =−γ+/summationtext∞ n=0(1/(n+ 1)−1/(k+n)) ; alsoY−n(z) = (−1)nYn(z). The Hankel functions are deﬁned to be H(1) ν(z) =Jν(z) +iYν(z), H(2) ν(z) =Jν(z)−iYν(z). (B. 129) The set {Jν,Yv}is a linearly independent pair of solutions of Bessel’s diﬀer- ential equation. The same is true of the pair/braceleftBig H(1) ν,H(2) ν/bracerightBig .TheWronskians are W{Jν(z),Yν(z)}=J/prime ν(z)Yν(z)−Jν(z)Y/prime ν(z) =2 πz(B. 130) and W/braceleftBig H(1) ν(z),H(2) ν(z)/bracerightBig =H(1)/prime ν(z)H(2) ν(z)−H(1) ν(z)H(2)/prime ν(z) =−4i πz.(B. 131) The functions Jν,Yν,H(1) ν,H(2) νall satisfy the same recurrence relations zFν−1(z) +zFν+1(z) = 2νFν(z) (B. 132) 2d dzFν(z) =Fν−1(z)−Fν+1(z) (B. 133) zd dzFν(z) =±νFν(z)∓zFν±1(z) (B. 134) d dz/bracketleftbig z±νFν(z)/bracketrightbig =±z±νFν∓1(z) (B. 135) ©200 1 CRC Press LLC and the diﬀerentiation formulae /parenleftbigg1 zd dz/parenrightbiggm/bracketleftbig z±νFν(z)/bracketrightbig = (±1)mz±ν−mFν∓m(z) (B. 136) dm dzmFν(z) =1 2mm/summationdisplay k=0(−1)k/parenleftbiggm k/parenrightbigg Fν−m+2k(z).(B. 137) In particular, J/prime 0=−J1, Y/prime 0=−Y1andH(i)/prime 0(z) =−H(i) 1(z),(i= 1,2). Thegenerating function is exp/bracketleftBig/parenleftbig t−t−1/parenrightbigz 2/bracketrightBig =∞/summationdisplay n=−∞tnJn(z) (B. 138) from which is derived cos (zsinθ) =J0(z) + 2∞/summationdisplay k=1J2k(z) cos(2kθ) (B. 139) sin (zsinθ) = 2∞/summationdisplay k=0J2k+1(z) sin{(2k+ 1)θ} (B. 140) cos (zcosθ) =J0(z) + 2∞/summationdisplay k=1(−1)kJ2k(z) cos(2kθ) (B. 141) sin (zcosθ) = 2∞/summationdisplay k=0(−1)kJ2k+1(z) cos(2k+ 1)θ (B. 142) Asymptotics. When |z| →0 withνﬁxed, the power series expansions (B. 126)–(B. 128) serve as asymptotic relations, Jν(z)∼/parenleftBigz 2/parenrightBigν 1 Γ(ν+ 1),ν/negationslash=−1,−2,... (B. 143) and when Re( ν)>0, Yν(z)∼ −iH(1) ν(z)∼iH(2) ν(z)∼ −1 πΓ(ν)/parenleftBigz 2/parenrightBig−ν . (B. 144) Whenzis ﬁxed and ν→ ∞ , Jν(z)∼1√ 2πν/parenleftBigez 2ν/parenrightBigν ,Yν(z)∼ −/radicalbigg 2 πν/parenleftBigez 2ν/parenrightBig−ν . (B. 145) ©200 1 CRC Press LLC Whenνis ﬁxed and |z| → ∞, Jν(z) =/radicalbigg 2 πz/braceleftbigg cos/parenleftbigg z−1 2νπ−1 4π/parenrightbigg +O/parenleftBig |z|−1/parenrightBig/bracerightbigg ,|argz|<π (B. 146) Yν(z) =/radicalbigg 2 πz/braceleftbigg sin/parenleftbigg z−1 2νπ−1 4π/parenrightbigg +O/parenleftBig |z|−1/parenrightBig/bracerightbigg ,|argz|<π (B. 147) H(1) ν(z)∼/radicalbigg 2 πzexp/bracketleftbigg i/parenleftbigg z−1 2νπ−1 4π/parenrightbigg/bracketrightbigg ,−π<argz<2π.(B. 148) B.5.1 Spherical Bessel functions The spherical Bessel functions jn,yn,h(1,2) nare deﬁned for integral nto be jn(z) =/radicalbiggπ 2zJn+1/2(z), yn(z) =/radicalbiggπ 2zYn+1/2(z), h(1,2) n(z) =/radicalbiggπ 2zH(1,2) n+1/2(z), (B. 149) and can be expressed in terms of elementary functions as jn(z) = (−z)n/parenleftbigg1 zd dz/parenrightbiggn/parenleftbiggsinz z/parenrightbigg , (B. 150) yn(z) =−(−z)n/parenleftbigg1 zd dz/parenrightbiggn/parenleftBigcosz z/parenrightBig . (B. 151) B.5.2 Modiﬁed Bessel functions Bessel functions with argument ±izare known as modiﬁed Bessel functions and are solutions of the diﬀerential equation z2d2w dz2+zdw dz−(z2+ν2)w= 0. (B. 152) The ﬁrst-kind and second-kind modiﬁed Bessel functions are deﬁned by Iν(z) =e−1 2νπiJν/parenleftBig ze1 2πi/parenrightBig ,−π<argz≤1 2π, (B. 153) Iν(z) =e3 2νπiJν/parenleftBig ze−3 2πi/parenrightBig ,1 2π<argz≤π, (B. 154) ©200 1 CRC Press LLC and Kν(z) =1 2πie1 2νπiH(1) ν/parenleftBig ze1 2πi/parenrightBig ,−π<argz≤1 2π, (B. 155) Kν(z) =−1 2πie−1 2νπiH(2) ν/parenleftBig ze−1 2πi/parenrightBig ,1 2π<argz≤π.(B. 156) Each is a regular function of zthroughout the z-plane cut along the negative real axis, and for ﬁxed z(/negationslash= 0) each is an entire function of ν; whenνis inte- gral,Iν(z) is an entire function of z. They constitute a linearly independent pair of solutions to the diﬀerential equation with Wronskian W{Iν(z),Kν(z)}=−1 z. (B. 157) Also Kν(z) =π 2 sin (νπ)[I−ν(z)−Iν(z)] (B. 158) where the right of this equation is replaced by its limiting value if νis an integer or zero. The series expansions are Iν(z) =∞/summationdisplay k=0(z/2)2k+ν k!Γ(k+ν+ 1), (B. 159) and Kn(z) = (−1)n+1In(z) ln/parenleftBigz 2/parenrightBig +1 2/parenleftBigz 2/parenrightBig−nn−1/summationdisplay k=0(n−k−1)! k!/parenleftBigz 2/parenrightBig2k + (−1)n1 2/parenleftBigz 2/parenrightBign∞/summationdisplay k=0[ψ(k+ 1) +ψ(n+k+ 1)] k!(n+k)!/parenleftBigz 2/parenrightBig2k ,(B. 160) whereψ(k) was deﬁned above. Also I−n(z) =In(z),K−ν(z) =Kν(z). (B. 161) Recurrence relations satisﬁed by modiﬁed Bessel functions include 2νIν(z) =zIν−1(z)−zIν+1(z), (B. 162) 2νKν(z) =−zKν−1(z) +zKν+1(z). (B. 163) Asymptotics. Whenνis ﬁxed and z→ ∞, Iν(z)∼1√ 2πzez∞/summationdisplay n=0(−1)n(2z)−nΓ(1 2+ν+n) n!Γ(1 2+ν−n),|argz|<π 2,(B. 164) and Kν(z)∼/radicalbiggπ 2ze−z∞/summationdisplay n=0(2z)−nΓ(1 2+ν+n) n!Γ(1 2+ν−n),|argz|<3π 2. (B. 165) ©200 1 CRC Press LLC B.6 The incomplete scalar product The incomplete scalar product for the family of Jacobi polynomials is de- ﬁned by Q(α,β) sn(t) =/integraldisplay1 t(1−x)α(1 +x)βP(α,β) s(x)P(α,β) n(x)dx, (B. 166) whilst its normalised counterpart is deﬁned by ˆQ(α,β) sn(t) =/integraldisplay1 t(1−x)α(1 +x)βˆP(α,β) s(x)ˆP(α,β) n(x)dx. (B. 167) Elementary properties of the normalised incomplete scalar product valid for alls,n= 0,1,...are ˆQ(α,β) sn(1) = 0, (B. 168) an index symmetry ˆQ(α,β) sn(t) =ˆQ(α,β) ns(t) , (B. 169) and ˆQ(α,β) sn(−t) =δsn−(−1)s−nˆQ(β,α) sn(t). (B. 170) Two other relationships frequently used are ˆQ(α,β) sn(t) =(1−t)α+1(1 +t)β [(s+α+ 1) (s+β)]1 2ˆP(α+1,β−1) s (t)ˆP(α,β) n (t) +/bracketleftbigg(n+α+ 1) (n+β) (s+α+ 1) (s+β)/bracketrightbigg1 2ˆQ(α+1,β−1) sn (t),(B. 171) valid when α>−1,β > 0,and ˆQ(α,β) sn(t) =−(1−t)α(1 +t)β+1 [(s+α) (s+β+ 1)]1 2ˆP(α−1,β+1) s (t)ˆP(α,β) n (t) +/bracketleftbigg(n+α) (n+β+ 1) (s+α) (s+β+ 1)/bracketrightbigg1 2ˆQ(α−1,β+1) sn (t),(B. 172) valid when α>0,β >−1.Formulae (B. 171) and (B. 172) are deduced from the relationships (1 .173) and (1 .174). Finally, the property ∞/summationdisplay l=0ˆQ(α,β) sl(t)ˆQ(α,β) l n(t) =ˆQ(α,β) sn(t) (B. 173) allows us to interpret the matrix operator K(t) with elements ˆQ(α,β) sn(t) as a projection operator on l2. ©200 1 CRC Press LLC Employ the following diﬀerentiation formulae, which follow from the index recurrence relations and the diﬀerential recurrence relations −d dz/bracketleftBig (1−z)α+1ˆP(α+1,β−1) n (z)/bracketrightBig =/radicalbig (n+α+ 1)(n+β)(1−z)αˆP(α,β) n(z),(B. 174) d dz/bracketleftBig (1 +z)β+1ˆP(α−1,β+1) n (z)/bracketrightBig =/radicalbig (n+β+ 1)(n+α)(1 +z)βˆP(α,β) n(z),(B. 175) and apply integration by parts to obtain two equivalent expressions for the incomplete scalar product, valid when s/negationslash=l, ˆQ(α,β) sl(z0) =(1−z0)α+1(1 +z0)β [(s+α+ 1)(s+β)−(l+α+ 1)(l+β)]× /braceleftBig/radicalbig (s+α+ 1)(s+β)ˆP(α+1,β−1) s (z0)ˆP(α,β) l(z0)− /radicalbig (l+α+ 1)(l+β)ˆP(α,β) s(z0)ˆP(α+1,β−1) l(z0)/bracerightBig (B. 176) and ˆQ(α,β) sl(z0) =−(1−z0)α(1 +z0)β+1 [(s+β+ 1)(s+α)−(l+β+ 1)(l+α)]× /braceleftBig/radicalbig (s+β+ 1)(s+α)ˆP(α−1,β+1) s (z0)ˆP(α,β) l(z0)− /radicalbig (l+β+ 1)(l+α)ˆP(α,β) s(z0)ˆP(α−1,β+1) l(z0)/bracerightBig .(B. 177) Thus, when s/negationslash=l,the incomplete scalar products ˆQ(α,β) sl(z0) may be calculated in terms of the normalized Jacobi polynomials ˆP(α,β) n. These polynomials are eﬃciently evaluated by a normalised form of the recurrence relation (B. 21) on the polynomial order: ˆP(α,β) n+1(x) = (ˆbn+xˆan)ˆP(α,β) n(x)−ˆcnˆP(α,β) n−1(x) (B. 178) with initialisation ˆP(α,β) 0(x) =/braceleftBig h(α,β) 0/bracerightBig−1 2, ˆP(α,β) 1(x) =1 2/braceleftBig h(α,β) 1/bracerightBig−1 2[α−β+x(α+β+ 2)]. ©200 1 CRC Press LLC The recurrence coeﬃcients are deﬁned by ˆan=an/parenleftBig h(α,β) n/h(α,β) n+1/parenrightBig1 2, (B. 179) ˆbn=bn/parenleftBig h(α,β) n/h(α,β) n+1/parenrightBig1 2, (B. 180) ˆcn=cn/parenleftBig h(α,β) n−1/h(α,β) n+1/parenrightBig1 2. (B. 181) The ratio of norm values in (B. 179), (B. 180), and (B. 181) are simple rational expressions in n,α, andβ; also ˆbn=(α2−β2)ˆan (2n+α+β)(2n+α+β+ 2). (B. 182) Whens=l,the following recurrence relation for the incomplete scalar product may be employed. Consider (B. 178) with n=sandn=l: ˆP(α,β) s+1(x) = (ˆbs+xˆas)ˆP(α,β) s(x)−ˆcsˆP(α,β) s−1(x), (B. 183) ˆP(α,β) l+1(x) = (ˆbl+xˆal)ˆP(α,β) l(x)−ˆclˆP(α,β) l−1(x). (B. 184) Multiply (B. 183) by ˆ alˆP(α,β) l(x), (B. 184) by ˆ asˆP(α,β) s(x) and subtract to eliminate the term containing x. Now multiply by the factor (1 −x)α(1 +x)β and integrate over ( z0,1) to deduce the following recurrence relation: ˆalˆQ(α,β) s+1,l(z0)−ˆasˆQ(α,β) l+1,s(z0) = (ˆalˆbs−ˆasˆbl)ˆQ(α,β) sl(z0)−ˆcsˆalˆQ(α,β) s−1,l(z0) + ˆclˆasˆQ(α,β) l−1,s(z0).(B. 185) Settings=l+ 1 in (B. 185) produces a recurrence formula involving ˆQ(α,β) ll, and three other incomplete scalar products of form ˆQ(α,β) nm withn/negationslash=m. ˆQ(α,β) l+1,l+1(z0) =ˆal ˆal+1ˆQ(α,β) l+2,l(z0) +/parenleftbigg ˆbl−ˆbl+1ˆal ˆal+1/parenrightbigg ˆQ(α,β) l+1,l(z0) + ˆcl+1ˆal ˆal+1ˆQ(α,β) ll(z0)−ˆclˆQ(α,β) l−1,l+1(z0).(B. 186) It may be initialised by the value ˆQ(α,β) 00(z0) =/braceleftBig h(α,β) 0/bracerightBig−1/integraldisplay1 z0(1−x)α(1 +x)βdx. (B. 187) ©200 1 CRC Press LLC Special cases commonly encountered are ˆQ(−1 2,−1 2) nm (cosθ0) =1 π/bracketleftbiggsin (n−m)θ0 n−m+sin (n+m)θ0 n+m/bracketrightbigg ,(B. 188) ˆQ(1 2,1 2) nm(cosθ0) =1 π/bracketleftbiggsin (n−m)θ0 n−m−sin (n+m)θ0 n+m/bracketrightbigg ,(B. 189) ˆQ(−1 2,1 2) nm (cosθ0) =1 π/bracketleftbiggsin (n−m)θ0 n−m+sin (n+m+ 1)θ0 n+m+ 1/bracketrightbigg , (B. 190) ˆQ(1 2,−1 2) n−1,m−1(cosθ0) =1 π/bracketleftbiggsin (n−m)θ0 n−m−sin (n+m+ 1)θ0 n+m+ 1/bracketrightbigg . (B. 191) These are valid when n/negationslash=m; whenn=m,the term sin (n−m)θ0 n−m occurring in (B. 188)–(B. 191) is replaced by θ0. ©200 1 CRC Press LLC Appendix C Elements of Functional Analysis C.1 Hilbert spaces In this section we collect some concepts from functional analysis. There are many standard introductory texts on this material, including [34, 33, 78, 10]. A Hilbert space is a vector space Hover a ﬁeld of either real or com- plex scalars, endowed with an inner product. The inner product is a bi- linear map that associates to each pair of elements f,ginHa complex number denoted ( f,g) with the following properties: (1) ( α1f1+α2f2,g) = α1(f1,g)+α2(f2,g) for allf1,f2,g∈H,and scalars α1,α2; (2) (f,g) =(g,f) for allf,g∈H,where the bar denotes complex conjugate; and (3) ( f,f)≥0 and (f,f) = 0 ⇔f= 0.We normally deal with real Hilbert spaces with a real inner product. The third property allows us to deﬁne the norm of an el- ementf∈Hto be /bardblf/bardbl= (f,f)1 2.It satisﬁes the properties (1) /bardblf/bardbl ≥ 0 and/bardblf/bardbl= 0⇔f= 0; (2) /bardblαf/bardbl=|α|/bardblf/bardblfor all scalars α; and (3) /bardblf+g/bardbl=/bardblf/bardbl+/bardblg/bardblfor allf,g∈H.Moreover, the Cauchy-Schwarz in- equality |(f,g)| ≤ /bardblf/bardbl/bardblg/bardblholds. The Hilbert space Hiscomplete with re- spect to this norm, i.e., every sequence {fn}∞ n=1inHthat is Cauchy (so that/bardblfn−fm/bardbl → 0 asn,m→ ∞ ) is also convergent to an element fofH (/bardblfn−f/bardbl →0 asn→ ∞ ). A basis for His a set of elements {e1,e2,...}ofHsuch that every element fofHis a unique linear combination of the basis elements: there exist scalars α1,α2,...such that f=/summationdisplay nαnen. (C. 1) If the basis can be ordered as a countably inﬁnite sequence {en}∞n=1His called separable, and the sum (C. 1) is interpreted to mean that /vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublef−N/summationdisplay n=1αnen/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble→0 asN→ ∞. (C. 2) (If the basis is not countable, then only countably many scalars in the sum (C. 1) may be nonzero and the sum is interpreted in the sense of (C. 2) for the nonzero scalar elements sequentially ordered.) The basis is orthogonal if ©200 1 CRC Press LLC (fn,fm) =hnδnm,wherehn=/bardblfn/bardbl2is necessarily positive. If hn= 1 for all n,the basis is orthonormal; this may always be arranged by replacing each basis element fnbyfn//bardblfn/bardbl. Example sofHilbertspaces. 1. Letl2denote the space of (real or complex) sequences {an}∞ n=1such that/summationtext∞ n=1|an|2converges. It is a Hilbert space with the inner product of sequences a={an}∞n=1andb={bn}∞n=1deﬁned to be (a,b) =∞/summationdisplay n=1anbn. (C. 3) An orthonormal basis is the set of sequences S={en,n= 1,2,...}where en={δnm}∞ m=1. 2. Letw={wn}∞ n=1be a positive real sequence, and deﬁne l2(w) to be space of (real or complex) sequences {an}∞ n=1such that/summationtext∞n=1wn|an|2 converges. It is a Hilbert space with the inner product of sequences a= {an}∞n=1andb={bn}∞n=1deﬁned to be (a,b) =∞/summationdisplay n=1wnanbn. (C. 4) The setSdeﬁned above is an orthogonal basis, and is orthonormal only if wn= 1 for alln.A particular example of interest is the choice wn=nµwhere µis a ﬁxed real number; this space is denoted l2(µ). 3. LetL2(a,b) denote the set of (real or complex) valued functions f deﬁned on the interval ( a,b) such that/integraltextb a|f|2converges. It is a separable Hilbert space with the inner product of functions f,gdeﬁned to be (f,g) =/integraldisplayb afg. (C. 5) The Lebesgue integral is used for this purpose with the understanding that two functions f,gare regarded as equal if they diﬀer only on a set of Lebesgue measure zero ( f,gare said to be equal almost everywhere ); this allows us to assert that the only function of norm zero is the function that is zero almosteverywhere. 4. Letwbe a real valued positive function deﬁned on ( a,b).LetL 2,w(a,b) denote the set of (real or complex) valued functions fdeﬁned on ( a,b) such that/integraltextb aw|f|2converges. It is a separable Hilbert space with the inner product of functions f,gdeﬁned to be (f,g) =/integraldisplayb awfg, (C. 6) ©200 1 CRC Press LLC with derived norm /bardblf/bardbl=/parenleftBigg/integraldisplayb aw|f|2/parenrightBigg1 2 . (C. 7) Ifαandβare real numbers exceeding −1, andwis deﬁned by w(x) = (1−x)α(1 +x)β,then the Jacobi polynomials/braceleftBig P(α,β) n/bracerightBig∞ n=1form an orthogo- nal basis for L2,w(−1,1),and the normalised Jacobi polynomials/braceleftBig ˆP(α,β) n/bracerightBig∞n=1 form an orthonormal basis. The cosine functions {cosnθ}∞ n=1and the com- plex exponential functions/braceleftbig einθ/bracerightbig∞ n=1form orthogonal bases for L2(0,π) and L2(0,2π), respectively. C.2 Operators A linear operator TonHis a function T:H→Hthat is linear: T(α1f1+α2f2) = α1T(f1) +α2T(f2) for allf1,f2,g∈H,and scalars α1,α2. Tis bounded if there exists a positive constant Msuch that /bardblT(f)/bardbl ≤M/bardblf/bardblfor allf∈H; the norm of the operator is then deﬁned to be /bardblT/bardbl= sup f/negationslash=0/bardblT(f)/bardbl /bardblf/bardbl= sup /bardblf/bardbl=1/bardblT(f)/bardbl. (C. 8) The null space N(T) ofTis the set {f∈H:T(f) = 0}; the range of Tis the imageT(H) ofHunder the action of T. An example is the integral operator Kformed from a real or complex valued kernel function kof two variables deﬁned on ( a,b)×(a,b) via K(f) (x) =/integraldisplayb ak(x,t)f(t)dt (C. 9) for each function f∈L2(a,b) ; the condition /integraldisplayb a/integraldisplayb a|k(x,t)|2dxdt< ∞ (C. 10) ensures that Kis a bounded linear operator on L2(a,b) with norm /bardblK/bardblnot exceeding/parenleftBig/integraltextb a/integraltextb a|k(x,t)|2dxdt/parenrightBig1 2.A discrete analogue is the operator Kwith associated matrix ( knm)∞ n,m=1deﬁned via (Ka)n=∞/summationdisplay m=1knmam,(m= 1,2,...), (C. 11) ©200 1 CRC Press LLC for each sequence {an}∞ n=1inl2; the condition ∞/summationdisplay m=1∞/summationdisplay n=1|knm|2<∞ (C. 12) ensures that Kis a bounded linear operator on l2with norm /bardblK/bardblnot exceed- ing/parenleftBig/summationtext∞m=1/summationtext∞n=1|knm|2/parenrightBig1 2. Of particular importance in numerical methods are projection operators P that may be characterised by the requirement that P2=P. In practice, such an operator is often associated with a ﬁnite dimensional space and is used to convert operator equations of the form Kf=gto systems of ﬁnitely many linear equations; the relation between the (computed) solution to the ﬁnite system and the original (inﬁnite dimensional) system is important in determining the success of numerical solution methods (see below). The adjoint K∗of a linear operator KonHis uniquely deﬁned by the requirement that (K∗f,g) = (f,Kg ) (C. 13) for allf,g∈H.The adjoint of the integral operator deﬁned in (C. 9) is an integral operator of the same form with kernel hdeﬁned by h(x,t) =k(t,x). (C. 14) The adjoint of the matrix operator deﬁned in (C. 11) is a matrix operator of the same form with matrix hdeﬁned by hnm=kmn, (C. 15) for alln,m = 1,2,... The operator KonHis compact (also called completely continuous) if for every bounded sequence {fn}∞ n=1inH,the image sequence {K(fn)}∞n=1has a convergent subsequence (in H). Bounded ﬁnite rank operators (those with ﬁnite dimensional range) are necessarily compact. The integral operator and matrix operator deﬁned by (C. 9) and (C. 11) are compact. By contrast, the identity operator Iis never compact in inﬁnite dimensional spaces. If {en}∞ n=1 is a basis for H,and{λn}∞ n=1is a sequence of scalars, the diagonal operator deﬁned by K(en) =λnen (C. 16) for allnis compact if and only if λn→0 asn→ ∞. Properties of compact operators are discussed in [34, 33]. In particular, the set of eigenvalues of a compact operator K(those values of λfor which the equation ( λI−K)x= 0 has nontrivial solutions x) is countable (perhaps ©200 1 CRC Press LLC ﬁnite or even empty); 0 is the only possible point of accumulation of this set. The Abel integral operator Adeﬁned onL2(0,1) by A(f) (x) =/integraldisplayx 0f(t)dt√ x2−t2, x∈(0,1) (C. 17) has norm /bardblA/bardbl=π 2and is not compact; for, as observed in [4], the functions fα(t) =tα(withα≥0),are eigenfunctions of AsatisfyingAfα=λαfα, where the eigenvalues λαvary continuously between 0 andπ 2asαranges from 0 to ∞,so thatAcannot be compact. The dimension of each eigenspace of Kis ﬁnite; for each λ/negationslash= 0,there is a unique smallest integer rso that the null spaces satisfy N((λI−K)r) =N/parenleftBig (λI−K)r+1/parenrightBig =N/parenleftBig (λI−K)r+1/parenrightBig =... (C. 18) and the range spaces satisfy (λI−K)rH= (λI−K)r+1H= (λI−K)r+1H=.... (C. 19) The spaceHhas the orthogonal decomposition H=N((λI−K)r)⊕(λI−K)rH (C. 20) (every element of His a unique sum of two orthogonal elements lying in N((λI−K)r) and (λI−K)rH). C.3 The Fredholm alternative and regularisation The following result, known as the Fredholm alternative , is very important in establishing the solubility of second-kind equations of the form ( λI−K)x= y,whereλis a scalar and Kis a compact operator on a Hilbert space H (λ−1Kis a compact perturbation of the identity operator I). We consider the four equations (λI−K)x=y (C. 21) (λI−K)x= 0 (C. 22) (λI−K∗)u=v (C. 23) (λI−K∗)u= 0 (C. 24) whereyandvare given elements of H. Theorem 7 (The Fredholm alternative.) The Equation (C. 21) has a solu- tionx∈Hif and only if (y,u) = 0 for all solutions uof the homogeneous ©200 1 CRC Press LLC Equation (C. 24). Thus if the zero solution u= 0 is the only solution of (C. 24), then for every y, the Equation (C. 21) is solvable, i.e., the range of λI−KisH;the solution xdepends continuously on y.Likewise, Equation (C. 23) has a solution u∈Hif and only if (x,v) = 0 for all solutions xof the homogeneous Equation (C. 22). Equations (C. 22) and (C. 24) have the same number of linearly independent solutions. These and allied properties of second-kind equations permit the construc- tion of relatively simple numerical methods that are stable and well-conditionedand for which error analyses are possible. Atkinson’s book [4] is a comprehen-sive survey of methods particularly appropriate to integral equations, payingattention to error estimates. In a similar way, Kantorovich [30] discusses errorestimates for second-kind matrix systems that are solved by the truncation method; Kress [33] also discusses such estimates in the context of projection methods. By contrast, ﬁrst-kind equations, such as Kx=y (C. 25) whereKis a compact operator (for example the matrix operator deﬁned by (C. 9) or the integral operator given by (C. 11)), are generally unstable,and simple numerical methods are ill-conditioned and yield poor results. Itis necessary to employ some method of regularising the equation. One suchmethod is Tikhonov regularisation, that consists of replacing (C. 25) by /parenleftbig ε 2I+K∗K/parenrightbig x=K∗y. (C. 26) For smallε, solutions to (C. 26) approximately equal those of (C. 25) (and are identical when ε= 0), but the precise selection of εis rather problem dependent and requires some care in achieving acceptably accurate numericalsolutions [22]. Many problems of diﬀraction theory and potential theory give rise to sys- tems of matrix equations or integral equations of the form Ax=y, (C. 27) which are singular in the sense that they are not of the second kind involving a compact operator. From a theoretical point of view it can be diﬃcult to establish whether such equations have solutions, even though there may begood physical reasons to expect the existence of a solution. Moreover, thecontinuous dependence of the solution xonyis not obvious, though clearly necessary for any physically plausible model of potential or diﬀraction. Froma computational point of view, the equation is likely to be unstable, i.e., small perturbations to yresult in large (and physically implausible) changes in the computed solution x.It is not diﬃcult to see how this eﬀect arises for the ﬁrst-kind Equation (C. 25) when the compact operator Kis given by (C. 16). ©200 1 CRC Press LLC Itisthereforedesirable,whereverpossible ,toconvertthesingularEquation (C.27)tooneofsecondkin dwithacompac toperatorforwhi chtheFredholm alternativeholdssothattheassociatedbeneﬁt sdescri bedaboveareobtained. Thisprocessi sknownas(analytical )regularisation.Itmaybedescribed formallyasfollows .Theboundedlinearoperator Riscalleda(left )regulariser ofAif RA=I−K whereKisacompac toperatoron H.Somegeneralpropertiesofregularisers aredescribe din[33].Applicationoftheregulariser Rto(C.27)producesan equationofth edesiredformat: (I−K)x=Ry. Ingeneral,theconstructionof Rmaybediﬃcult,ifnotimpossible.How- ever,thedualseriesequation sarisingfromth epotentialproblem sanddiﬀrac- tionproblemsconsideredinthisbookan ditscompanionvolumeca nindeed beregularised ;theregularisationprocessisexplicitlydescribe dinSection2.1, although the regulariser appears only implicitly in the analytical treatment of the dual series equations. The regularised equations enjoy all the advantages of second-kind equations for which the Fredholm alternative holds, includ-ing precise estimates of the error or diﬀerence of any solution computed to atruncated system, from the true solution (as a function of truncation numberN tr). The error decays to zero as Ntr→ ∞ (and in practice quite rapidly beyond a certain cutoﬀ point, usually related to the electrical size of the body in diﬀraction problems). The same remarks apply to triple series equations, as well as to the dual and triple integral equations arising from the mixed boundary value problems associated with Laplace’s equation, the Helmholtz equation, and Maxwell’sequations for the various canonical structures described in these volumes. ©200 1 CRC Press LLC Appendix D Transforms and Integration of Series D.1 Fourier and Hankel transforms The Fourier transform of the function fdeﬁned on ( −∞,∞) is F(y) =/integraldisplay∞ −∞f(x)e−2πixydx, (D. 1) and its inverse is given by f(x) =/integraldisplay∞ −∞F(y)e2πixydy. (D. 2) Precise conditions on the validity of the inversion formula is given in [9]; a particular useful class for which it holds is Lp(−∞,∞) with 1 ≤p≤2. The Hankel transform of the function fdeﬁned on (0 ,∞) is F(y) =/integraldisplay∞ 0Jν(xy)f(x) (xy)1 2dx, (D. 3) and its inverse is given by f(x) =/integraldisplay∞ 0Jν(xy)F(y) (xy)1 2dy. (D. 4) The inversion formula is valid for parameter ν/greaterorequalslant−1 2whenfis integrable on (0,∞) and of bounded variation near the point x,and is continuous at x; if fhas a jump discontinuity at x,the left-hand side of (D. 4) is replaced by 1 2(f(x+ 0) +f(x−0)) (see [61]). D.2 Integration of series In this section we present some results on the validity of term-by-term integration of series. ©200 1 CRC Press LLC Theorem 8 Let{fn}∞ n=1be a sequence in L2(a,b),converging to fin the L2norm, i.e., /bardblf−fn/bardbl=/parenleftBigg/integraldisplayb a|f−fn|2/parenrightBigg1 2 →0,asn→ ∞. Letgbe a function in L2(a,b)and deﬁne h(x) =/integraldisplayx afg, h n(x) =/integraldisplayx afng. Thenhnconverges uniformly to hon[a,b]. Proof Fixx∈[a,b] ;from the Cauchy-Schwarz inequality, /parenleftbigg/integraldisplayx a|f−fn||g|/parenrightbigg2 ≤/integraldisplayx a|f−fn|2/integraldisplayx a|g|2. LetA= 1 +/integraltextb a|g|2.Then, given ε>0,there exists Nsuch that when n>N, /integraldisplayb a|f−fn|2<ε2/A,so that/integraldisplayx a|f−fn||g|<ε. Thus,hnconverges uniformly to hon[a,b]. Corollary Let/summationtext∞ n=1fnbe a series with fn∈L2(a,b)and converging to f in theL2norm, i.e., /vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublef−n/summationdisplay r=1fr/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble= /integraldisplay b a/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglef−n/summationdisplay r=1fr/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 1 2 →0,asn→ ∞. Then the series∞/summationdisplay n=1/integraldisplayx afng is uniformly convergent to/integraltextx afgon[a,b]. In particular, the Fourier series of any function in L2(a,b) can be integrated term-by-term over the interval [ a,x]. The series/summationtext∞ n=1anof real terms is Abel-summable if lim r→1−0∞/summationdisplay n=1anrn ©200 1 CRC Press LLC exists. The series/summationtext∞ n=1fnof real valued functions on [ a,b] isuniformly Abel- summable on [a,b] to the function f,if for allε>0,there is some δ>0 such that for all x∈[a,b], /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay n=1fn(x)rn−f(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle<ε for 1−δ<r< 1. For each ﬁxed rwith 0<r < 1,the power series/summationtext∞ n=1fn(x)rnis uniformly convergent on [ a,b] to its sum, and may be integrated term by term. It imme- diately follows that term-by-term integration of a uniformly Abel-summable series is justiﬁed. ©200 1 CRC Press LLC References [1] Abramowitz, M. and Stegun, I.A., Handbook of Mathematical Functions , Dover, (1965). [2] Akhiezer, N.I. and Glazman, I.M., Theory of Linear Operators in Hilbert Space , Vols. 1 & 2, Pitman Publishing Co. 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