paper on mehler integrals

ON A GENERALIZATION OF MERLER'S INVERSIONFORMULA AND SOME OF ITS APPLICATIONSbyPAUL LESTER ROSENTHAL A THESISsubmitted toOREGON STATE UN VERSITY in partial fulfillment ofthe requirements for thedegree ofDOCTOR OF PHILOSOPJune 1961 APPROVED: zvProfessor of MathematicIn Charge ot Major bbliiman ot ieparment of Mathematics Date thesis is presented MAY).961Typed by 141NPotter Redacted for Privacy Redacted for Privacy Redacted for Privacy Redacted for Privacy ACKNOWLEDGMENTThe author thanks Dr. F. Oberhettinger for hissuggestions and direction during the development of thematerial in this thesis. TABLE OF CONTENTSPage 401FOR THE FUNCTION1.I**S00THE GENERALIZED SPHERICAL WAVEREPRESENTED AS A GENERALIZEDMEBLER TRANSFORM.24BIBLIOGRAPHY *..*.32APPENDIX . ***S*34ChapterIINTRODUCTION *2INVERSION FORMUP-Ii+ix(Y)I by?4eblEON A GENERALIZATION OF MERLER'S INVMRSION FORMULA AND SOME OF ITS APPLICATIONS INTRODUCTION ie equivalent pair G(y) F(x)tauoh(Trroposed to generalize an inversion torsiulainvolving conical functions. g()9+) f(x)00f(x)z tanh(x) P(y) g(y)dy g(y) HereP.14(y)4enot.s the first Legendre functionof order -*A + ix withy1 (8, Ch.).Theseconicalfunction occur in certain boundary valueproblems involving configurations of a conical shape.The paper by Mohler referredto before deals with thedistribution of electrostatic charges on an infiniteconical shell.In a following paper, Neumann (19) ccs.tended Mehler's investigations to the case of the dis-tribution of charges under the influence of an external field.The analysis for the derivation of (a) as usedby Mohler is purely formal.The first reliable investi-gation as to the class of functions f( x)for which (a)holds, seems to be due to Pock (13). In recent timesconsiderable interest in inversion formulas of thetype (a) has been displayed. For instance, Karp (4)gave the charge distribution on a finite conical shell(cup) employing the Wiener-Hopf technique; Carslaw (5),Buchholz (2, 3, 4), Felsen (11, 12), among others, ivestigates the propagation of (plane and sphericalelectromagnetic and acoustic waves in a (infinitelytended) conical horm.It will be shown in section 2 that an inversionformula similar to (a) can be given when, instead ofp..16+ix(y)p.,,16+ix(y)is used.This inversion formulareduces to (a) for the special case /Is O. The proofof this inversion theorem is based on the analysis emg*ployed by Pock (13).It may be mentioned in this con-nection that a recent survey (14)1 concerning integraltransforms makes no mention of Mehler's formula.1Shortly after writing this thesis, the author discoveredthat a generalization of Mehler's inversion formulafor the caseitss qa, where n1,2,3,etc. had beentreated before (21) Section 3 will represent the general additiontheorem for the modified Henkel function of the Gegen-bauer type (9, p. 43) in the form of such a generalizedMohler transform (generalized spherical wave) Finallypreviously known results as integral expressions for vhecylindrical and spherical wave (9, P. 55) and certainintegral representations for the product of two modifiedHankel functions (.0., 7) can be derived as special cases. (0.2) (0.3The integral expression for the conical function156) fi7cosht)gcos(xt)dRel-b=147='1-0suggest instead of (a) the nveroformula g(Y)INVERSIW FORMULAOR THE FUNCTIONP+ixJw1 1"(31- 09f(dcos(xt)dt, II 7ra-(7rx) fl*u+iflOILL+00fg_r241.or the equivalent pa (0.4) (0.5) (07)00fir(2)/ i(x) . Tr- by (8, p.50) ag cosh(7I-1(y) one obtains from (0.3) for the special caserespectively putting ycosh a00(sinh a) g(cosh a)jrf(x)cos(xa)dx(0.600)1 (sinhcosh a)cos(xa)da0(axa) VIcosh where(g)(2)t(g)7r-inh ag coshin cosh 0/1)d/I nhA-/10Thesek4.6)and (07)]are the Fourier eosinand the Fourier sine transform formula. ThereforeFouriert s inversion formulas are a special ease of thegeneralized Mehler transform.Equation (0.3) will now be proved. TheoremIf a function Vgx) given in the interval/-Q8 1x-coo) is such that O(t) s (Binh t/2) (cash t)has its first derivative integrable oven an infiniteinterval (0= t00 ) while its second derivative isabsolutely integrable overUrn95(t)motRz5.0,)(t)0 be integrable for Rept.4,0and listcb(t):0,then Ox) is representable in the formt-4.00of the integral r (Yr-001 pigosh 0)9/(cosh0).sinh 0 d0te interval, and if Theorema function f(4) is absolutely integrableover any finite interval and f(4) 0(j/.42-13-(7)/1-1.00 has its derivative absolutely integrable overany finite interval, and if f(0) = 0, then f(g) isrepresentable in the form of the integral (2), whereOcosh 0) is defined by 1)The proof of theorems 1and 2 will now be given.The proof of theorems 1 and 2konsists in examining the course of computations which6ormally led to the fora of the inversion integral (2)when given integral (1).Starting with where(1.2) gy/(cosh 0) cosh a 0 cosh e so (1.2) becomes(cosh(g)41 -Y2+cosh a(M)[1-10.-ki. 0ya6, cosh x is cosh Replacing Rhs (right hand sidegives1.4) which is (1.4)cosh 0)(7/(3111{[Areq3).]4*00feeILLcosh 0- coshdt'Aukt(cosh e)06.7rsinh0"colit (coeb-cosh t)0Rep 0Assume next f(g)Er such that order of ints-) into (1 1) gration can be interchanged in (1.4). But the conditionsstated in theorem 2 allow this since this gives rise toabsolute integrability of (1.4) so by Fubini's theoreminterchange is allowed.So (1.4) now becomes(1.5)t1I(cosh0)p777)sinh e)1,r.)] (cosh. coshlit f(kOtttildt0Next letscosh 0, Mrcosh tso (1.5) now becomes so.6) become (1.7) 0[(cosh)j./.])(v20But (1.7) is in the form of an Abel integral equation.The conditions of Theorem 1 are such that (1.7) can besolved and (1.7) becomes (1.8)c-11'0h Q1-1Icos[icosh-141](s -)-hf()*mI dgTrikin(h+077]vf(co.ir _cr) -1h+p(cr2_1)-P/2dCf where-36h.With cosh-ls . 0 (1.8) becomesor2 (1.9)001 )sin [0/2+07] e cos(eg)f(g )(VI(27r)r0cosh QOCr)(cosh eCf) Since the conditions of Theorem 1 and 2 are such thatthe inverse Fourier integral exists, it follows from(1.9) (1.10)f(il(2/7r)3 2cos/Ler ye--)sin [0i+fivr]cosh0V.1(0')(cosh00'2/2e0'de Integrating (1.10) by parts (this is permittedsince by Theorem 19I(X) is such that the operationis valid) gives(0' 21)/2-0/2dcf10 (1.11)f(g)(2/7r)3/2I"(cosh G[cosge()(cosh 8-Cr) Letcosh Q1(do, (1.12)g)=Yi(CrgeoshCr)0' 2P/2/Then (1.11) becomes by Theorem 1 and fact that lowerlimit is zero by definition of the improper integral](Cr2-1)/2.4. g19 . 000cosh QIgsineper) (cosh0(CI 2--P/2 clGr desin[(iii+p)nl a 14)f(g)(2/7)312 r(y2-osin07T1g.coshjisin(g)ji.VACr( osh 19-C (0/ 2-1)3413 a de Integrating over a triangular domainobserving that the conditions of Theorem I are suchthat Fubini s theorem is valid, (1.14) becomes 15)f(g.(2/7r Pliccf)2 r( cr213/2 cosh tdtsincosh-1ge12 (cosh e-(1)-3/24.1j. de dCfi hCr0, cosh t (1.16)f g)(2/7r)/ rOsin[(+)7r] g00sinh t-0+1/sinALt But (1.17)iyosh a) r-s-iy)sinh 'Try (sinh00sin xycosh x - cosh a) .= Re /I .=1(20p. 165).Letat, y =y2-p,= e.Then By (1.18) and (1.19), (1.16) becomes13 (1.18)(ashricfig) r0-p-ig ) sini2Trg (sinne)(27n F(413)00isinge(cosh 8 - cosh dO.tAlso (1.19)7t[0407r[Fo64-o(r2-3) (1.21)h(t)Ilfr(cosh e)(si0coshd Proof:By the following transformationscosh 6 .cfcosh tsCr. (s-1)w+ 1and use of hyperbolic identifies (1.21) becomes(1.20)1.(117rg siI"([34-ikt )*fgPP(cosh e)tif(cosh e)ig -0sinh e de. Herepcan be any complex number such that Rhs of(1.20) is well defined.It will be next shown how the conditions imposedonyrof Theorem I came about.From (1.12) the following lemma is used to meetthe condition thatg(e)0as eIf lim 0(cosh e)0, where0(cosh e) a- 00YAcosh 6)sinh1-241/2,then lim h(t) r 0, where (1.22)h(t)0 using next a mean value theorem or improper in aa(1.22) becomes (1.23)h(t)1-21.2(s11)+1] 2 w,0where 0 .= I) .= 1.For (1.23) tto exist 3iPalso if 033..= 1 (s-1)w+2 -f31(s-1)-k-13wheres1so1.23)becomes (1.24) /h(t)/Z 21 213 /1(s-1)/9+1.7/1-a--w5413 dw. But(1-36-13dwis the beta function B(4-36-13 36.4.p) and therefore the above is valid for all p such thatp(-0h-13,3+13) exists.Thush(t) -.. 0astcoby given in lemma. This gives rise to one ofhe conditions required ofis Theorem 1.Going from steps (1.9) to (1.10) requiredthatbe such that this is permitted. This requiresthat dit/dt is absolutelyintegrable in (0,co ) and d2h/dt2absolutely integrable in (0,a) 0 L azco.Equation (1.23) can be written as follows: (1.25)h(s)1-2p0[(p +1] dw, 17/0?0 where again a mean value theorem for improper inteis used, or )7(s).21-4 01-(s-lr1,7g)108-1) gj5ig)So from (1.25)dh/dtd2h/dt2depends directly ondO/dtandd2 0/dt2respectively.Hence assumedO/dtis absolutely integrable in Z0,C0> andd 0/dt2is absolutely integrable in4'0,a?a> 0andarbitrary.Going from steps (1.9) to1.10) is nowvalid.The condition thatm 0( )0, Re B0 rest+0from the following lemma: where .28)0Y(cosh e).[FP0Using1.2) and fact(cosh 0cosh t(cosh ecosh t po.e is such thatcosh ez 2,tio4:ot using also osh e)2/fl-)jrcosh e - cosh td (cosh(9/2)li(tanh2) p.173) )complete elliptic integral of first kind and(cosh 9/2)-1sinh 9/2)-1gives(cosh)/Z/f(y2...9 (si9/2 )1/This proves the lemma.The condition thatfca)0, /1 m 0resultsfrom observing formula (2).Reversing the steps tobtain (1) requires that f(fl)meet the conditions ofFourie is inversion theorem (see steps (1.10) and (1.9)).17o h e2)near 18The integrability condition forf(//)is due to asymptotic behavior of the kernel (the LegendreFunction of the first kind) This completesthe proof of Theorems 1 and 2.As an example for the transforms (1) and (2), thetranerormqabwill be computed to give af.Then thetransform of thisfwill be computed to recoverLet(2.1)(cosh 09)sin a (cosh 0 + cos/2+P(sinh where .--/TzEtz17Re P4,/_ 3.meets theconditions of TheoremInserting (2.1) into (1) and using ) gives (2.3)si/Tx/ ) 7-106p+i/i/ )/(7T/20 /10acosh 49cosz). o hcos8.)-3/2+an 9 (sinh 0)de. Rewriting2.3co sa (cosh e - cosh t (2.6) is the transform fr,17113 ad19 (2.4)(I/77- -1 Ilisinh( /7/i) f(y2) co (cosh 0 + cos3/213 de dt. Letcosh 0 - cosh t(cosh t + cosz, (2.4) becomfcli) .71-1Si/TA )RY2-13+iA)) (772) a cos(// t)sin a (cosh t + cos coz(1+z)/2 4*dz.0Both integrals are known.(2.6) fy2-3/2--/-70).7-1 /012.1.0., )I(*i3-i/u ) B O).)sinh aA7 f(/// )/abut sincea r (I/YPfsca)/dyt"-for all0 0such that 0 4 Sofcif )meets requirements ofTheorem 2.Now taking (2.6) and computing the inversetransform of (2.6)(2.1) will be obtained.Using (1.1) and (1.17) gives (2.7))b(coshe)0/7() sinh(/7,sith,/sinp t) cosh tcoshat 71-177,-////bi)(hr/2) 710/2'.`P,1)sinh,udit Rewriting2.7)20 0.0f1(cosh t - cosh 6)cofsin(/'t) /t/ail :I1 :1Ian ( sidfidoBut(2.9)fsia///t sinh(0sin a acos a + cosh t)Proof of (2.9) thus gives (2.10)t(cosh)e) (2/1)71(f3)C (re-13)7.-"(coshtcosh e0ssinh ta + cosh2 ay With the substitutioncosh t - cosh e(cos acosh )z(2.10) become21(2.8)cosh 6)r 77-/ 2)(-13, gF.10-Z/-101 (0,1T(5T'F)deo] 04,0()ii = (ci (d-a/V t_go-501(1+0,./.2Y(T41-) AIMBOO + V00)US BOO + V SO("za)(equTB)7 ((eLisoo) zp(Z+T) +soo)(e IISOO +SOO) V uTe 0Ta) t-LI30 .1(.1/a)1(0T2TS); //CC g-e7i)Evv.(e/iittsoo) Then (2.12) becomes (2.16)Y(coshe) .sin acos aco-3/2+ (2.16is (21as was to be shown.23 3.THE GENERALIZED SPHERICAL WAVEREPRESENTED AS A GENERALIZED MEHLER TRANSFORMA basic solution of the modified wave equation2Af -cfm.0 in 2n + 2dimensions is17)3.1)fHere Kn(oR) is the modified Hankel function of order(9, Ch. 7) where R is defined(3,2)B14a2+b22ab cos AFor the two or three dimensional space (o. a 0 or nrespectively)freduces to a cylindrical wave Ko(24 or to a spherical wave Rexp (-cR)respectively.An expression for (3.1) has been given by Gegenbauer(9, p. 43) in the form of a series representation knownas the (Gegenbauer) addition theorem of the modifiedRanks], function(3.3)K (R) ab) r(v)aband for a b the same formula with a and b interchanged,It will be shown later that instead of (3,3) theowing relation can be proved.riss0cos A) ( .4) (.5)2ab cosA) +24000si(Tr 40r(v+r(viix)Kx) -cos A) dx, +2ab cos (absin fl(v+ix) ['(v .ix) cos B) 41x,xvka201 0 -=A27r Here.3444x-cos A) is the Legendre function ncut" (see Appendix c)$ p143) (note that (3.4) issymmetrical inaandb which3.3) is not).PutATrBand get Re B.:4111 7T25 sinh (7Y) t follows fromp. 122equation (7)that))141() (3.6) F(v+ix) r(ix) -A+ixRe vObviously the integral3.6) is of the type of aMohler transform (0.3).But the integral in (3.6)also be regarded as a Lebedev transform 9. p* 75)obtain hen instead of3.5)+2abz)-* Xvka 4-b2+2abz 2)677-:"3/2(in case y26 sinh (7rx)s a one valued function of sin the complexz planecut along the real s axis from 4.1 *4.00thereforewe choose in3.5) cos Bss we can writ Special CasesThe express on (3.4) can be used to obtainpressions for the cylindrical (v * 0) and spherical wave=in the form of an integral expression.a) qy indrical xml Cv =0)Since (8. p150) (3.8)(a +obtains from.4) -2ab cos ACoos 0)(,vr)0008+3091 0(b)herical wavevSince (9, p. 10)yro7r/2)one obtains-2ab cos00)(a)bath (gz)-3+ix008A)dx.27 The expressions.7) and8were previously kno(9, p. 55)If we apply the inversion formula (0.3) for (3.6)we obtain 00(3.9)x(a)ixb(TT/2)ab)v)((72- 2aby)If1/4v/iab2+2aby /61 P h+ix(7) 4YThis formula is valid for Re (8, p. 163), andseems to be new.(For formulas ofhis type see (6 and7).)The special case v0givesc,0(3.10)()Kix(b)f KI-. a2v. o0observing that (8, p, ) fia2 dt.+2abt)+2ab cosh P/( osh t)(3671)(sinh t)cosh/r(vOi)t7The casev r 14gives(4.0)Kix) K( b)14/7- Cab)28 Equivalent with (3.4) is(3,11)( 2 b2-2ab cos11(y) 12)-.v -1xK4ka2 b2-2ab cosA)} cos A ab, Re irY aHere dr-cos A) is the Gege bauer fction175) A Inserting this into (4.1) and using the relation (9, p.5)x(a)i 277sinh ('Ti)(a)one finds ( .4).Proof of ()0Consider the integral in1) taken oclosed contour in the complex x plane consisting ofthe real axis between Rand +Rand the semicircle29 of radiusR in the upper half plane.ChooseRsuch, that the semicircle separatesconsecutive poles xit,i(v+// )and/4/44st i(v4v/ +1)of the integrand..The residue ofthe integrand of (3011) at a pole x/0(v+// ) isequal to( 1)+// )A)(a)m 0, It 2).The contribution of the integration along thesemicircle tends to zero when R , provided a>This follows from the behavior of the integramd in(3.11) on the semicircle for large values of RAppendix a)We obtain therefore by the residue theoremfrom (3.11) a +-2ab 008 A)Xviia2 b cos A%I* 2 p. 176)A)(-1)//C,(cos A).(right band. side) of the last ezpresson is30 equal to the Rhs of (3.3This proves3.4) and31 32 BIBLIOGRAPHYBromwich, T. 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Akademiia Nauk (Doklady) SSSRLenim-rad 118:219-222. 1958. it follows that in (3.11)) (71/ )04 ria Zi+0(1./// both expressions valid in 71- erg /1/4/7- and in the second expression /a not an integer. ano. APP In order to show that the integral in (3.11) along a semicircle of radius R in the upper x'-halt plane tends to zero as R ' 1"°4D, it is necessary to investigate the integrand of (3,11) for x R 0_4 c,e/r From Stirling's formula (8, p. 47) and the definition of the modified Bessel functions (9, p. 5) one obtains for fixed s and large (complex) /1t or Yi+o(141/ 17 2) co**// lon-fi r substituting e/2 e //(log/' 1) 34 (cosy )()77Bin,inf06vfor large/vin thealt plane (corre -cosA).7.6.77, (s)//c sZ-fi A+,6 /7 71+0(1P .)]forargez\inZrarg), 4/7Therefore, because of (3.12) with X la+0 1./A for large /(/in-77-argfizThe integrand in (3.11) is therefore equal toOffi***I( 011))**1(b)Behavior of1)//(z)for x-Yr+ixIt follows from (8p. 129, formula6)) that ,evoCe ))735 ponding to the upperx half planeFurthermorefrom8, p. 147ormu a (5))together with Stirling's formula Berinition of(z)$ not aoe Legendre tunction (8), 1.-*/u theabetween 1-/./ I36