lame by computer

LAME FUNCTIONS OF THE FIRST KIND GENERATED BY éCOMPUTER z H.G.WALTER “ European Space Operations Centre ofESRO (ESOC), Darmstadt, Federal Republic ofGermany (Received 1May, 1970) Abstract. The definitions oftheLamé functions and theellipsoidal coordinates inherent tothem are introduced. Proceeding from Lamé’s differential equation thefour classes ofLamé functions ofthe first kind aregenerated bycomputer with theaidofformula manipulation techniques. For this pur- pose algebraic expressions forthecoefficients intheLamé polynomials areconstructed byvirtue of recurrence formulae and presented intabular aswell asmachine readable form forfurther processing. The determination ofaparameter appearing inthepolynomial coefficients leads toanalgebraic eigenvalue problem. Itisshown how this eigenvalue problem isamenable tonumerical treatment. The polynomial coefficients together with theeigenvalues arecharacteristic fortheLamé functions and define them completely. 1.Introduction Inconnection with theoretical investigations, theLamé functions have been repeatedly employed fortheformulation ofproperties emerging frombodiesofellipsoidal shape. Historically, Lamé had introduced these functions inconjunction with theproblem of determining thesteady temperature inanellipsoidal conductor when thetemperature isprescribed ontheboundary oftheconductor. Ingeophysics and particularly in thearea ofequilibrium figures, occasionally theLamé functions have been used when ahigh degree ofprecision isimposed onthemathematical formulation ofstructures (Oppenheim, 1922; Darwin, 1910). More recently they have become amenable tothe representation ofthepotential oftheEarth aswell asother celestial bodies (Dou- bochine, 1966). Asystematic account ofLamé’s investigation was given byHeine (1878) and subsequently byHobson (1931). Considerable use ofthis account has been made inparts ofthis paper. Itisarequirement ofthe practical applications that theLamé functions aremade available inamachine readable form forfurther processing bymeans ofacomputer rather than inatabular compilation. Therefore theapproach pursued inthis report aims atthegeneration ofthecharacteristic coefficients oftheLamé polynomials, which aretheprincipal constituents ofthefour classes ofLamé functions ofthefirst kind. Once these coefficients have been established, itisonly amatter offormality tobuild upthepertinent Lamé functions. The line ofthought governing thegeneration ofLamé functions starts outfrom a differential equation ofthesecond order usually referred toasLamé’s equation. In terms ofindeterminate coefficients, thestructure ofthepolynomial expressions which form part ofthesolutions ofthisdifferential equation areknown from thetheory. Byinserting them into thedifferential equation, recurrence formulae forthedeter- Celestial Mechanics 4(1971) 15-30. AllRights ReservedCopyright ©1971byD.ReidelPublishing Company, Dordrecht- Holland ©Kluwer Academic Publishers *Provided bytheNASA Astrophysics Data System : 16 H.G.WALTER é mination ofthepolynomial coefficients areobtained through asetofconditional A equations. Inasecond step, therecurrence formulae areapplied toestablish theA complete expressions oftheindividual coefficients uptoanyspecified order. . Because oftheformal character oftheapproach, itexposes itselftotheutilization offormula manipulation techniques. Indeed both theformation oftheconditional equations and thegeneration ofthepolynomial coefficients forthefour classes of Lamé functions have been accomplished bymeans ofFORMAC (FORmula MAnip- ulation Compiler). More precisely the computer programs have been written in PL/I-FORMAC (IBM, 1967) and allowance hasbeen made topunch thepolynomial coefficients asPL/I statements forprocessing intheframework ofrelated investiga- tions. With only little modification, the coefficients can bepunched inaformat compatible with thedefinition ofFORTRAN statements. Thispaperiscomprised ofthreechapters.Inthecourseofthefirstonethedefinition oftheLamé functions isgiven jointly with theadopted notations and only asmuch ofitispresented asisessential foranunderstanding oftheensuing paragraphs. The second chapter deals with theformal solution ofLamé’s differential equation resulting intheconditional equations and recurrence formulae forthepolynomial coefficients. Finally, thethird chapter isdedicated tothedetermination ofthepolynomial coef- ficients, which leads toanalgebraic eigenvalue problem. The method foritssolution isbriefly described. Forillustration andchecking purposes, aselection ofpolynomial coefficients pertaining tolower order Lamé functions ispresented intheAppendix. 2,The Definition ofthe Lamé Functions Linked totheLamé functions aretheellipsoidal coordinates which aretheindependent variables ofthis type offunction. The relationship between rectangular coordinates x,¥,zand ellipsoidal coordinates £,,£2,3isdescribed inthesubsequent paragraph followed byaparagraph onthevarious classes ofLamé functions. Wherever no misunderstanding can arise thesymbol ¢stands foreach ofthethree ellipsoidal coordinates. 2.1 ELLIPSOIDAL COORDINATES Forgiven rectangular coordinates x,y,zthecubic equation in6, 2 2 2 ey z ete Feo" o with 0<h<k, hasthree realroots ¢7,€3,€3,which fulfill theinequality 0<G<K<G<k<ti<o. Q) Onsubstituting these three roots into (1), theequations ofthree confocal surfaces aredefined intherectangular coordinates x,y,z:anellipsoid, ahyperboloid ofone sheet and ahyperboloid oftwo sheets. Ifthethree equations resulting from (1)by inserting 2,€3,€3aresuccessively solved forx*,y*,27,oneobtains forthecoor- ©Kluwer Academic Publishers +Provided bytheNASA Astrophysics Data System LAME FUNCTIONS OFTHE FIRST KIND GENERATED BYCOMPUTER, 17 i dinates ofthepoint ofintersection thefollowing relations: j daeés eae ) peGol)G=i)Oe~8) ® i?(i?—h) 2 eV eee 2GoW =a e-8) 6(2) Itisinthissense that Heine (1878) and Hobson (1931) have formulated thedefinition oftheellipsoidal coordinates €,,€),€;ofapoint interms ofitsrectangular coor- dinates x,y,z. Byvirtue of(3), (4), and (5)aunique relationship between therectangular coor- dinates and theellipsoidal coordinates isestablished, provided that anappropriate convention ontheselection ofthesign has been adopted when thesquare root is taken ontheright hand side ofthepreceding equations. 2.2 THE FOUR CLASSES OFLAME FUNCTIONS OFTHE FIRST KIND Intheframework ofthis report thesymbol E(é) isrepresentative ofany Lamé function ofthefirst kind. Especially £;(¢), s=0, 1,...,2n,denotes thesetofLamé functions oftheorder n.They aresubdivided into four classes which foragiven n encompass 2n+1linearindependent functions. Let an forneven o= )4(n—1) fornodd. Then theclasses arerepresented by o+1 functions Ks(2) n—o functions 15,(é) n—o functions Ms(2) «—functions NS(é), and K,L,M,Nareabbreviations fortheexpressions defining theLamé functions of thefirst, second, third and fourth class respectively: a,forneven Ki©=agtt+ayset?ong$08 1 (6)=ag.8"+an {formosa, ) BAG)=JE=B[bog"4bye? bp-o-s.s€ forneven ofc fornodd ® ©Kluwer Academic Publishers *Provided bytheNASA Astrophysics Data System 18 HG. WALTER i Mi()=J@-# [asfeetoz Cpenta€ fornever+{ert forea|® MQ=JP—0 P=[oerdyetto d,-;,, forneven oF.:10) frree|(9) Assoon asthepolynomial coefficients 4,s,bj»,Crs»d,sin(7)to(10) aredetermined, the Lamé functions ofthe four classes will becompletely defined. Inthe ensuing sections the algebraic procedures for the determination ofthese coefficients are supplied. 3.The Generation ofthe Lamé Functions 3.1 THE CONDITIONAL EQUATIONS FOR THE POLYNOMIAL COEFFICIENTS Inthetheory oftheLamé functions (Hobson, 1931) ithasbeen proven that E(é) is thesolution ofthedifferential equation @E dE 2nay py yeeee (1)(E~)est(2E e +{p(+k)—n(n+1)@}E=0, (il) wherepisaparameter, which willbedefined later.Forthesakeofconvenience and better visualization thesubscripts nand superscripts shave been deleted here. They will also beomitted insubsequent occurrences ofLamé functions wherever mis- understandings cannot possibly arise. From thepreceding section itisinferred that thepolynomial coefficients inthe expressions KO)=Pk(®), (12) L@=Ve-" AO, (13) M()= JP —FPu), (14) N@=JE =JF Py) (as) must bedetermined sothat thedifferential Equation (11) issatisfied by(12) to(15). Pk;Pi,Pyand Py,areabbreviations forthepolynomial parts inEquations (7)to(10) respectively, and aresometimes called Lamé polynomials. Substituting successively theright hand sides of(12) to(15) and their first and second derivatives into (11) yields thefollowing differential equations forPx,Pi,Pu and Py. >dP, dP, 2yey Pegcag—pe—eyshe (P=)(@~)Ger+ERE Ee +{p(P+e)—n(n 41E}Pe=0 (16) ©Kluwer Academic Publishers *Provided bytheNASA Astrophysics Data System LAME FUNCTIONS OFTHE FIRST KIND GENERATED BYCOMPUTER 19 j which isevidently identical withthedifferential Equation (11)fortheLamé functions. : Furthermore, oneobtains é PP, dP,e 2_2)(€?—k?)SE+€(40?—n?—3k?) + &VEdaetetas le +{p(h? +k?) —k?—(n—1)(n+2)@}P,=0, (17) WP, dP, 242)(@2_gaydPu, 22342) (=WE~1)Gat+O48 Ve +{p(h? +) —h? —(n—1)(n +2) 2}Py=0, (18) whichresultsfrom(17)whenAisreplaced byk, Py dP, 2_py(@2—payPsy3¢(222—2—eyPv &VE daetBCE ae +{(p— 1)(#?+) —(n(n +1)—6)7}Py=0. (19) The next objective isthedetermination ofthecoefficients inthepolynomials Px,P,, Pyand Py.Let a=W +k, B=hk?. (20) Onsubstituting thepolynomial Pxand itsfirst and second derivatives in(16) one finds theconditional equations forthepolynomial coefficients byequating tozero thecoefficients ofthevarious powers of&,namely 2(2n—1)a,—a(p—(n—0)*) ay=0 4(2n—3)a,—a(p—(n—1))a;—Bn(n—1)a9=0 6(2n —5)ay—«(p—(n~4)*)aa—B(n—2)(n—3)ay=0 20(2n+1—20)a,—a(p—(n+2—20)*) a,_y— —B(n+4—20)(n +3—20)a,-.=0 (20+2)(2n=1=20)a,4, —«(p—(n-20)") a,— —B(n+2-20)(n+1-20)a,_,=0. (21) Here and inthefollowing paragraphs thesecond index shasbeen deleted wherever theclarity oftherepresentation isnotaffected. Likewise theconditional equations forthepolynomial coefficients inP,,Py, Py are deduced. 3.2 THE RECURRENCE FORMULAE FOR THE POLYNOMIAL COEFFICENTS Itbecomes evident from theconditional equations that thepolynomial coefficients a; ofPxcan bedetermined bymeans ofarecurrence formula, whichisoftheform 2i(2n +1—2i)a,=a{p—(n+2-2i)P} ay+ +B(n+4-2i)(n+3-2ia2, 22) a_,=0, ay=C,=constant, i=1, 2,...,¢+1. ©Kluwer Academic Publishers *Provided bytheNASA Astrophysics Data System 20 ‘H.G.WALTER g Analogously bysubstitution ofP,,Py,Pyintothecorresponding differential Equations : (17),(18),(19)recurrence formulae forthedetermination ofthepolynomial coefficients = b,ofP,,¢,ofPyand d,ofPyareobtained inturn: 2i(2n+1—2i)by=[a{p—(n+1—217}- —(2n+3-41) k?]by,+B(n+3—2i)(n+2—2i)b,-2, (23) b_,=0, bp=C,=constant; i=1, 2,...,(n—o). Onreplacing kbyAin(23)therecurrence formula forthecoefficients c;ofPy takes the form 2i(2n +1—2i)¢=[a{p —(n+1-2i)?} — =(2n+3—4i) h?]c-;+B(n+3—-2i)(n+2—-2i)c-2, (24) ¢-1=0, co=C3=constant; i=1, 2,...,(no), and finally one finds forthecoefficients d,ofPy: 2i(2n +1-21) d;=a{p—(n+1—2i)} dy + +B(n4+2-2i)(n+1-2i)d)2, (25) d_,=0, dy=C4=constant; i=1, 2,...,0. Theconstants C,(j=1, 2,3,4)may bechosen arbitrarily, e.g.C)=1. Each ofthe recurrence formulae contains theparameter p.Inorder tocomplete theconstruction ofthe polynomial coefficients the parameter pmust bechosen according tothe constraints towhich itisexposed. This will bethesubject ofthenext chapter. 3.3 FORMAC APPLICATIONS The conditional equations forthepolynomial coefficients inPx,P;,Pye,Pyhave been constructed byusing the algebraic manipulation features ofthe programming lan- guage PL/I-FORMAC (IBM, 1967). Moreover, validity checks have been performed confirming theformation ofthe recurrence formulae (22) to(25). After theverification ofthe recurrence formulae they were transcribed into the FORMAC language resulting inprograms which basically allow thealgebraic formulation ofthepoly- nomial coefficients uptoanyarbitrary order. Precautions were taken intheprograms sothat thegenerated coefficients can subsequently beutilized asinput tofurther studies. Afew sets ofpolynomial coefficients oflower order arelisted forreference inthe Appendix. Infact, thecoefficients uptotheorder n=20 could easily begenerated on anIBM computer 360 Model 50.The increasing length oftheresulting expressions made itimpractical togenerate thecoefficients ofstill higher orders than 20besides thefact that limitations ofthecomputer configuration prevented it. 4,Determination ofthePolynomial Coefficients 4,1 THE PARAMETER CONTRAINTS Inorder toarrive atinteger rational functions forPx,P,,Py,Pyonemust make the ©Kluwer Academic Publishers *Provided bytheNASA Astrophysics Data System 4 [LAMEFUNCTIONS OFTHEFIRSTKINDGENERATED BYCOMPUTER a é recurrence formulae (22) to(25) subject toconstraint which demands that thepara- 3 meter pinthepolynomial coefficients must bedetermined sothat . Gg41=Ae41,g=0 inPx(S)=Px (26) Dy-o=bn-o,s=9 inPL(E)= Py (27) Cra=Cros=OinPy(E)=Py (28) d,= dy,,=0 inPy(é)=Py. (29) Ifthecondition (26) isintroduced intheconditional Equations (21) forthepolynomial coefficients inPxasystem of+1 linear homogeneous equations intheunknowns dp,dy,...,d,isobtained, Putting d=ap (30) the matrix Ux=(im) G1) of(¢+1)-th order pertaining tothesystem iscomposed oftheelements listed below: Uyy=yg=o yg =0, uy,¢=2(2n—1), uyerr=a? A; Ug,=Uz2== 2,92 =0, Uz,9-1=4(2n —3), us,=a(n— 2-4, Uz,a41 =—Bn(n—1); Usy=sq ="=3,43 =0, Us,¢~2 =6(2n— 5), u3,g-1 =a(n—4) -A, Us,q=— B(n—2)(n—3), Ms,041 =05 Year =a(n—2oP—2, Us+1,2 =—B(n+2—20)(n+1—20), Uos1,3 =Maria == Wester =0. Essentially Uxisafunction ofp,A,andk.Anecessary andsufficient condition fora non trivial solution ao,dy,...,a,oftheaforementioned system oflinear homogeneous equations is detUx=0. 2) ©Kluwer Academic Publishers *Provided bytheNASA Astrophysics Data System 22 +H.G.WALTER j Inananalogous manner theconditional equations fortheparameters passociated 3 with P,,PyandPycanbeestablished; i.e. 5 detU, =0, U;,of(n—o)-th order (33) detUy=0, Uyof(n—@)-th order (34) detUy=0, Uyof-th order. (35) Ithasbeen proven inthetheoretical treatment oftheLamé functions (Heine, 1878) that foreach value n=0, 1,2,...there exist ¢+1 real and unequal values forp associated with Px; n—o real and unequal values forpassociated with P,;n—o real and unequal values forpassociated with Py;¢realandunequalvaluesforpassociated with Py, adding uptoatotal of2n+1 values corresponding tothe2n+1 linear independent Lamé functions oftheorder n. Inorder tofind these distinct values ofptheconditions (32) to(35) aresubjected toappropriate matrix transformations which lead toalgebraic eigenvalue problems. 4.2. THE EIGENVALUE PROBLEM Ifthem-thcolumn ofthematrix Uxisinterchanged withthe(«+2—m)-th one,where ‘massumes thevalues m=, 2,...,¢+1, then one obtains det(Ty—AI)=(—1)1?%* detUg. (36) Herein, Istands forthe unity matrix and Txforatriple diagonal matrix whose elements arerelated tothose ofUxthrough bho42-m =Wim+Ae, (e) 1=1,2,...,¢4+1;m=1,2,...,6415 0for I1#o+2-m e= 1for [=o+2—m. Ascan beinferred from (36) thefulfilment ofthecondition det Ux=0 isequivalent with thedetermination oftheeigenvalues ofthematrix Tywhich isoftriple diagonal form: thy typ O 0 DO oe ee 0 0 tay tea ty 0 0 HO 0 Pe 0 .G8) On ) fette letter. According totheproperties oftheLaméfunctions theeigenvalues underconsideration areallreal and unequal. Consequently there exist ¢+1eigenvalues which byvirtue ofEquation (30) yield ¢+1different values fortheparameters p.Thus with reference to(12) the¢+1 Lamé functions ofclass Kareestablished assoon astheeigenvalues ©Kluwer Academic Publishers *Provided bytheNASA Astrophysics Data System {LAME FUNCTIONS OFTHE FIRST KIND GENERATED BYCOMPUTER 2B é have been calculated. Since inthepast numerical analysis combined with computer 3 usagehavemadethecalculation ofeigenvalues amanageable task, theaforementioned z eigenvalue problem canbeconsidered assolved. Inthiscontext itmay bereferenced . that theleft-right-algorithm developed byRutishauser (1958) was used inseveral test cases forthenumerical calculation oftheeigenvalues andhasproven tobeanefficient method. Inananalogous manner theparameters pertaining totheother classes ofLamé functions canbefound bysolving theeigenvalue problems described below. The functions oftheclass Lareamenable totheeigenvalue problem det(T,—Al)=(—1)"2@-9- 9-9) detU,=0, (39) Nyn-o+-m =Mim+48, (40) 121, 2, 2-0; m=1, 2, 03 0for l#n-c+1-—m e= 1for l=n-o+1-m. Aneigenvalue problem formally identical with theone fortheclass Lfunctions holds fortheclass Mfunctions except that kisreplaced by/twherever itappears explicitly. The functions ofclass NVareassociated with theeigenvalue problem det(Ty—AI)=(—1)" detUy=0, (41) fio+1—m =Mim+4°6, (42) 121, 2,240; m=1, 2,05 0for I¢o+1—m PSU forl=o+1—m. The preceding four eigenvalue problems give rise tothedetermination of (9+ 1)+(n—o)+(n-c)+o=2n+1 (43) values oftheparameters p.Because of(12) to(15) thedefinition of2n+1 linear independent Lamé functions oforder ncanthus beconsidered accomplished. 5.Conclusion From theactual generation oftheLamé functions emerged thedistinct impression that much oftheir complexity hasbeen removed bytheutilization ofcomputerized formula manipulation techniques. Hence, thelengthy and cumbersome expressions forthe Lamé functions should nolonger restrain their applications tocelestial mechanics, geophysics and other appropriate domains. The simplifications aremainly ascribed tothealgebraic formulation oftheLamé functions inamachine readable form forfurther algebraic and numerical treatment. Also, theusual apprehension connected with thealgebraic eigenvalue problem embedded inthetheory oftheLamé ©Kluwer Academic Publishers *Provided bytheNASA Astrophysics Data System 4 H.G.WALTER i functions hasexperienced remarkable attenuation bypowerful procedures foreigen- 3 value calculations being practiced nowadays. Inview ofthecapabilities ofmodern é computer systems, theeigenvalue problem and themathematically involved expres- . sions fortheLamé functions donotanylonger impair their useinanalytical and numerical investigations. Byway ofillustration some samples oflower order Lamé polynomial coefficients arecompiled intheAppendix. Shortage ofspace prevents theinclusion ofthehigher order coefficients because oftheway they increase inlength, and, besides, they areof nofundamental importance inthiscontext, However, theauthor will bepleased to reproduce these and higher order coefficients inmachine readable form should itbe desired. Acknowledgements This work materialized during theauthor’s stay with IBM Cambridge intheautumn of1969. The author wishes toexpress hisdeep gratitude totheIBM Corporation which partially sponsored theproject. Heisparticularly indebted toJ.P.Rossoni, Head ofthe Cambridge Advanced Systems Department, and his staff’ members P.Sconzo and D.Valenzuela forthenumerous fruitful discussions. Itwasexceedingly instructive totranscribe themathematical formulations into theFORMAC language under theauspices ofD.Valenzuela. Likewise theauthor appreciates thecomputer services ofthe IBM-Boston Programming Center. Last but not least hewishes to give distinct credit toIBM Germany and theEuropean Space Research Organization (ESRO) forthesupport received inevery respect. References Darwin, G.H.:1910, Scientific Papers bySirGeorge Howard Darwin, Vol. IIL,Figures ofEquilibrium ofRotating Liquid and Geophysical Investigations, Cambridge University Press, Cambridge, pp. 186-316. Doubochine, G.N.: 1966, ‘Sur ledéveloppement dupotentiel delaterre parlesfonctions deLamé’, Trajectories ofArtificial Celestial Bodies (ed. by J.Kovalevsky), Springer Verlag, Berlin, pp. 68-74. Heine, E.: 1878, Handbuch der Kugelfunctionen, 2nd edition, Vol. 1,Theorie der Kugelfunctionen, Berlin, republished byPhysica Verlag, Wiirzburg, 1961. Hobson,E.W.:1931,TheTheoryofSphericalandEllipsoidal Harmonics, Cambridge University Press, Cambridge, republished byChelsea Publishing Company, New York, 1955. IBM: 1967, PL/I-FORMAC Interpreter User's Reference Manual, 360 D03.3.004, IBM-Boston Programming Center. Oppenheim,S.:1922,DieTheoriederGleichgewichtsfiguren derHimmelskérper, Encyklopidie der Math. Wissenschaft., Vol. VI,2,B.G.Teubner, Leipzig, pp. 1-79. Rutishauser, H.: 1958, ‘Solution ofEigenvalue Problems with theL-R Transformation’, US Bur. ‘Stand. Appl. Math. Ser. 49,pp.47-81. Appendix ALGEBRAIC EXPRESSIONS FOR THE LAME POLYNOMIAL COEFFICIENTS Below, thepolynomial coefficients ofthefour classes ofLamé functions ofthefirst ©Kluwer Academic Publishers +Provided bytheNASA Astrophysics Data System : LsNcronso ReKRnaacor x g kind arecompiled uptotheorder n=6. Thenotations used comply with those : introduced inSection 3with theunderstanding that 5 ALPHA stands foro, BETA stands forf, Pstands forp, subscripts areplaced between parentheses. The computer output format has lent itself directly tothedisplay ofthegenerated expressions. 1.Polynomial Coefficients oftheLamé Functions ofType K N=0 1=0 A()=C N=1 I=0 A()=C N=2. 1=0 AQ)=C I=1 A(1) =1/6PALPHA A(0)—2/3ALPHA A(0) N=3 I=0 A()=C I=1 A(1) =1/10PALPHA A(0) —9/10 ALPHA A(0) N=4 I=0 A()=C ©Kluwer Academic Publishers *Provided bytheNASA Astrophysics Data System : 26 HG.WALTER j I=1 é A(1)=1/14PALPHAA(0)—8/7ALPHAA(0) I=2 A(2) =3/5BETA A(0) —1/14 PALPHA? A(0) + +1/280 P?ALPHA? A(0)+8/35ALPHA? A(0) N=5 1=0 A(0)=C I=1 A(1)=1/18PALPHA A(0)—25/18ALPHA A(0) 1=2 A(2)=5/7BETAA(0) —17/252 PALPHA? A(0)+ +1/504P?ALPHA? A(0)+25/56ALPHA? A(0) N=6 1=0 A()=C I=1 A(1)=1/22PALPHA A(0) —18/11 ALPHA A(0) I=2 A(2)=5/6BETAA(0) —13/198 PALPHA? A(0)+ +1/792P?ALPHA? A(0)+8/11ALPHA? A(0) 1=3 A(3)=13/396 PBETA ALPHAA(0)—379/693 BETAALPHAA(0)++7/297PALPHA® A(0) —1/594 P?ALPHA? A(0)+ +1/33264 P?ALPHA A(0)—16/231 ALPHA?A (0) 2.Polynomial Coefficients oftheLamé Functions ofType L Nel T=0 B()=C ©Kluwer Academic Publishers *Provided bytheNASA Astrophysics Data System : LAMEFUNCTIONS OFTHEFIRSTKINDGENERATED BYCOMPUTER 27 j N=2 i 1=0 . B(O)=C N=3 I=0 B(0)=C I=1 B(1)=1/10PALPHA B(0) —2/5ALPHA B(0)—1/2K?B(0) N=4 1=0 B(0)=C I=1 B(1) =1/14PALPHA B(0)—9/14 ALPHA B(0) 1/2 K?B(0) N=5 I=0 B(0)=C I=1 B(1)=1/18PALPHA B(0) —8/9ALPHA B(0)—1/2K?B(0) I=2 B(2) =—1/36PK?ALPHA B(0)+29/126 K?ALPHA B(0)+ +3/7BETA B(0)—5/126 PALPHA? B(0)+1/504P? ALPHA? B(0)+ +8/63ALPHA? B(0)+5/56K*B(0) N=6 I=0 B(O)=C I=1 ©Kluwer Academic Publishers *Provided bytheNASA Astrophysics Data System : 28 +H.G.WALTER g B(1)=1/22PALPHA B(0)—25/22ALPHA B(0)—1/2K?B(0) i I=2 . B(2) =—1/44PK?ALPHA B(0)+137/396 K?ALPHA B(0) + +5/9BETAB(0)—17/396PALPHA? B(0)+1/792P?ALPHA?B (0)++25/88ALPHA? B(0) +7/72 K*B(0) 3.Polynomial Coefficients oftheLamé Functions ofType M N=1 1=0 c)=c N=2 I=0 C=C N=3 1=0 C@=Cc I=1 C(1) =1/10 PALPHA C(0) —2/5ALPHA C(0) —1/2H?C(0) N=4 I=0 C=C I=1 C(1) =1/14PALPHA C(0)—9/14ALPHA C(0)~1/2H?C(0) N=5 I=0 C=C I=1 ©Kluwer Academic Publishers *Provided bytheNASA Astrophysics Data System : LAMEFUNCTIONSOFTHEFIRSTKINDGENERATEDBYCOMPUTER 29 j C(1)=1/18PALPHA C(0)—8/9ALPHA C(0)—1/2H?C0) Q I=2 . C(2)=—1/36PH?ALPHA C(0)+29/126H?ALPHAC(0)+ +3/TBETA C(0)—5/126 PALPHA? C(0)+1/504P? ALPHA? C(0) + +8/63 ALPHA? C(0) +5/56 H*C(0) N=6 I=0 C@=c I=1 C(1) =1/22 PALPHA C(0) —25/22 ALPHA C(0)—1/2H?C(0) I=2 C(2)=—1/44PH?ALPHA C(0) +137/396 H?ALPHA C(0) + +[5/9 BETA C(0)—17/396 PALPHA? C(0)+1/792 P?ALPHA?C (0)+ +25/88 ALPHA? C(0) +7/72 H*C(0) 4.Polynomial Coefficients oftheLamé Functions ofType N N=2 I=0 D(0)=C N=3 I=0 D(0)=C N=4 1=0 D(0)=C I=1 D(1) =1/14 PALPHA D(0)—9/14 ALPHA D(0) N=5 1=0 ©Kluwer Academic Publishers +Provided bytheNASA Astrophysics Data System : 30, HG,WALTER j D(0)=C i I=1 ‘ D(1)=1/18PALPHAD(0)—8/9ALPHA D(0) N=6 1=0 D()=C I=1 D(1)=1/22PALPHA D(0)—25/22ALPHA D(0) I=2 D(2) =1/3BETA D(0)—17/396 PALPHA? D(0)+ +1/192P?ALPHA? D(0)+25/88ALPHA? D(0) ©Kluwer Academic Publishers +Provided bytheNASA Astrophysics Data System