nasim 650256

Internat. J. Math. §Math. Set m VoL. 789. i(198i) 171-180, THE MEHLER-FOCK TRANSFORM OF GENERAL ORDER AND ARBITRARY INDEX AND ITS INVERSION CYRIL NASIM Departsent of Yatheaatics and Statistics The University of Calgary Calgary, Alberta, Canada (Received July 24, 1983) ABSTRACT. Anintegral transform Snvolving the associated Legendre function ofzero order, Pgyz.(2)s =€[1s0), astheKernel (considered asafunction of),1scalled enler-Fock transform. Sonegeneralizations, involving thefunction 7,2), vhere have been known for some tine. Inthis preseat note, vedefine ageneral Nehler-Fock transform involving, a8theKernel, theLegendre function P,,(2), ofgeneral order and anarbitrary index -24 t, t=4ix,-© <1<, Then wedevelop asymetric fnversion formulae for these transforms. Many well-known result are derived as special cates ofuMs general fors. These transforas are widely used for solving many axisymetric potential probless. KEY WORDS AND PHRASES. Mehtor-Fook tranefom, Kontorovich-Lebedev traneforme, Legendre functions offiret and second kinds, Macdonald function, Bessel functions. 1980 MATHEWATICS SUBIECT CLASSIFICATION CODE. 44418 1, twropvcrroN. Tosolve many axisyemetric potential problens, integral transforms involving associated Legendre functions as kernels, are used widely. When the associated Legendre function ofthefirst Kind, Pg,),(2), #€(ly), 1sused asthekernel, the transform iscalled Nehler-Fock transform oforder zero, (1, p.175(8,9)], and index soregeneralassoctated Legendrefunction,nasely,Fy,;,(2),myanon-negative Anceger (2,p.390] andH4,,(2)s where w#0[3],havebeenknow forsometine, [ef 4]. tatt these generalizations, the emphasis has been togeneralize the order uoftheLegendre function F(z), butnoattempt hasbeenmade toextend theresults for anarbitrary index v. Inthis note wedefine transforms, involving anore general form ofthe kernel function, the associated Legendre function ofthe first kindPA4(2),vithindex ¢=o+fx,-e<1<8anduanarbitrary complex number. fninversion foraula 1sdeveloped, also, involving theLegendre function Ps) as a c. NASIM kernel, thus establishing asymmetric inversion theory for general Mehler-Fock trans~ formation. Also, @Farseval type relation for these general transforms isproduced. ALL the well-known results are derived as special cases. 2. THE PRELIMINARY RESULTS. We note below, for future reference, sone of the properties of the associated Legendre function ofthefirst kind, P!(2) oforder uandindex vforunrestricted vu and #[1,111]. The function isone-valued and regular inthe complex plane supposed cut along the real axis from 1to -*. Pay =Pye) en =EQ) peg Hie)=FD ew, 1,250 (2.2) ‘An Important case iswhen y= 0and v4san non-negative integer. Then pe - @y",Pat ae the Legendre polynomial. Now, consider the representation, (1, p.128 (28)], ryt gor Pooay =24(a2-ayFYGHED ory[-vu,$s1-20;EEE ¥ TG z eae Res >LandRey<1, Also, [2,p.78(1)], sora) «P20) faigead Fadio® =TareeareD |,TaTae 1 :.fawe0 Getrs)® Reo >Reb>0. Using the inequality r 1os, ee, Q-tta)? -2)* it 4s clear that aFy(a,byoia) s2(as) Thus2 2"(aavgP=iyHt a Pia) = ir: Reu<}eras) 2?) a1) whence, Pl@y~a¥ as +e (2.3) and Pe~pF ase. 2.4) ‘Also, that 2FyoHtr,bies2) =0G), as|r]+9, whence PesG@)=00) as|r]+, 2.5) 1 providedReu<3. MENLER-FOCK TRANSFORM AND ITS. INVERSTON wn We shall also need the following result, (5, p.75(76)]. Lemna 1. Let$0) =00), asve =00%), asv+0. 7 1pote 1 sa)-2[ temocnae, a>=F, 2.6 trenott=[L2zenav, an 0 where K,(0) andZ,(0) aretheMacdonald function andtheBessel function respectively oforder t,acomplex number. Next let usconsider the contour integral re[+xcoocae . where C1atheclosed contour inthes-plane asshow. NowX,(u) {8anentire func~ tion of2and 41sclear that g(a) t= x regular inside and on Cfrom (2.7) above. Thus, byCauchy's eheoren ofresidues, Tso, : ; wots otter onsfTCetipWoa(orenay : m9 ae 1 : ‘ 2 z +f cert Kpnaterirnde 2, ot +8]CHDRoseyacrtindy +f)cto&,,,tag@inae =htheyen, oy, Now the existance ofthe integral in(2.6), implies that l@HnkK,We@Hin| +0 as|r]+, thus lg] and [Z,[both vanish as|x]+=. Aad we now have, Ty+-t;0|e]+= #fGoHtKypyCoatorenrdy =-t[Come Ryyoottvrdy onsimplifying, weobtain ote ote [oSsecoaeae =-fyeMaaads +ifW, using (2.6) andthefact that Kj) =KW). am c. Nas Thus, 1pteseo=Ayfpaale(9C@)92)Ide (2.8) From the definition ofthe function ggiven by(2.7), g(a)-g(-n)=fLYU,-Liev,0 ° If we let = gla) = gt-2)-FAGZEC®)«r.2),say, then (2.8) and (2.9) reduce to :te soy=&fs stncoark,cortaresand o-t= re)=[Lxeora, 0 respectively. Now we have ageneralized Kontorovich-Lebedev transforms inasymmetrical form, (ef. 6]. Thus, we have shown: Lema 3.Letf(v) =0(e™”), asus =00, asvo. a 2on fo=Bf" ssinoareoreras , (2.10)then ote re)=[EDners, ean were [ol<3. If we set ¢=0, the above pair reduces to fo-%ii+sinh(or)k(PCED where ren=fLOx,corde, ° ie siving usthe usual Kontorovich-Lebedev transformation (2, p-361]. Lema 4,Tfy(t)€L(o-t=, ote), ol<4,andRey<1, thea ; ; - ey ette otis¢ 7 -fat es iGore C.vcor4,arate=/2C.WOKae- Proof. Consider the double integral [fe POscent, ,rat||2Ge Jonge (OMe MENLER-FOCK TRANSFORM AND TTS INVERSION 175 ajotte)syeC) < gy|edy Wes|Gaye Me FL eewortetycree[eed LL. Ga? rere se _ “Jflslacay+f a eei leae —y ~~ é : wf é : =voting" acdy+[P|2, oreo|aray ff aflworolas fe aya w(otin lar+ . ayeah Iam 8yay? f|y(otiz) |dr<=,duetothehypotheses. Hence rpmscopes1docteGane? Pyewhaeay iisabsolutely convergent, and therefore the change oforder ofintegration ispossible. And, oy pheleo dyIHPerat ove oy y =feae| I. 1gape Heay ote=fork "veongterde, kev<a. ote byevaluating the y-integral, (6,p-179(1)]. 3. The main results and special cases. Theorem 1. Letfy) =0(e%), assy >= =o), asyolt " oteé 1 1 , On Fsin(st)P =v=TG =u+OPH WP(eae cnet) then : FQ)»i"SUPE dy 6.2) wheret=o+it, -e<r<m, lol<$-u, Rev<d. Proof. Using theestimates (2.3)and(2.4)ofthefunctfon P,,,(y), along withthe conditions imposed onf,the integral in(3.2) defining the function g,exists. Note that inorder toensure the existence ofthe integral in(3.1), representing the func~ tion f,onemust have t!”*4F(t) €L(o-i=,oti=) atleast. Now let, WoFry ord- veore ox then from (3.1) 176 c.nasi 2pe 5m= C#sin(rer(erPey, de« (3.4) Next, wedefine anintegral operator. of=1, nef Ee fundneve feniseosyssls] Eerea, new. Note that this 1saLinear, self adjoint operator, asort ofgeneralized incomp- lete Laplace operator. Now applying this operator toboth sides ofthe equation (3.4) above, weobtain fa rns £[<=afo DPEud cfd «$f ©sim(ne9(e)-P (urde 1gay Gaye dete wee : ote bf ftsmomsernycerae, due to lemma 4, or, = ivo:ff 1GannFO=ae]#aimeeconntordt, whence, according tolena 3,wehave the inversion, 1=[2gerd ,oP provided the integral exists. Also, Lf got of2% 00+Ffnow av[7Ge an ) — eo +Fftaae[oho Kerner« wp0 Thechange oforder ofintegration canbejustified duetoabsolute convergence, by making useoftheestinates oftheMacdonald's function X,(¥) along with thecondi- tons tnposed onthe function f. Now the v-integral can beevaluated, (6,p-198(27)], togive, . o@)«FG-u-org-u+o fFOP dy« 1 And from (3.3), wehave finally, Pepe ren=[OP eae. as required. Thus wehave #ayenetric transformation, inthat, if pte 1 1 x to+f" ecsincord -y-ord-»+oFrod,then ote reo=frrr cna, defining amore general form ofthe Nehler-Fock transform oforder yand arbitrary NENLER-FOCK TRANSFORM. AND TTS. INVERSION 7 Corollary: Ifthe function f,satisfies the conditions ofthe above theorem, thea Ef fecineord=vordv4OP,coetdsenda foray] J)esr—word-u+oPanek,00ronduae.6.5) Note that from the definition ofg, in(3.3), wehave g(-t) =g(t) (3.6) Now we shall look atsome ofthe special cases. Let =0, Then (3.1) reduces to fy)=ah|bsin =u=Ord=utOP@prcendt r : cop :“2inrsinh(rord -u-eord- w+EOP) Pend. ence sq)=f sincord -y-dord -y+tory,wrdnar FJ, z 2 Sate , using the properties (2.1) and (2.6), where (it) = MeineP(it)Jireeectnae, giving usageneralized Mehler-Fock transformation [3]. Further {fwe put y= 0, then the above pair 8 reduced to = [contre yy, WFtindr sw)iENP4(WP): and. Fen=[rere (a the usual Mohler-Fock transformation ofzero order (2, p-389]. Next, we shall derive, formally, aParseval type relation. Let us define the function F(t) and G(t) te be the generalized Mehler-Fock transforms ofthe function f(y) and g(y) respectively, so chat P(t)=ffreeora anda ace)=ifBWP dy, where ¢=0+tt,|o|<4,Rey<1and-»<r<=,Andofcourse, asproved above jotta 1 1 roZe mer -u-ord=v+oP,rede, and asimilar formula for 9(y)- Consider the integral, 2pte 1 2EJ +imag -v-org=u+wetoceae ore potion PEL veorcoemat, vay 18 cc. Nasi Elcorcme [ow, na2)te” 7OOPOG ie ate&]eweCOPerPydy +fowrure .1 Thus we have, Theoren 2. let Fand bethe generalized Mehler-Fock transforms offand g. Then, _joe 1 ‘[Te sincord -u-ord-u+rwmowrae +fsemowa on 1 This is«formal derivation, but the analysis can bejustified byabsolute convergence, using suitable conditions onthe functions savolved. Ifveset c= 0in(3.7), wecan easily deduce that 2ff<stmcrord -v-eord~u+enreenaenae °=fraw . os)1 aknown result (3]. If, further, weput u=0,then (3.8) reduces to (2,p.394), Aff«cannesnecenaeena: =[reneunay « where, - F(it)-fryt.1POP gov andasimilarrepresentation forG(éz). Although the functions Pand fsatisfying (3.1) and (3.2) are defined for restricted values of 4,one can formally extend the results as follows: we ape 1 af=J(ebainGrG -m-Orm+OPMWPedt, —(3.9) then wie P(E)=ffOF, ay @.10) 1Ope Od where t=04it,Jol<4,we<rewandm=01,2... «And alltheresults derived above hold. For instance, ifo=0,then vehave from above, 1 : =rd-m+ensey=cayFecannery)EA neti where a rd+m+in Fen=[0% (04- MENLER-FOCK TRANSFORM. AND TTS. INVERSTON 179 TEweter rGmein4 =——— ran, rd+n+in MeoefroyETE .es|LOreneingeas FO=[,s@PheerWey using the property (2.2), where s@)=CD)i,xcanFW (Odt giving usapair ofMehler-Fock transform oforder m,[2, p-416]. Itmay bepointed out here that one can prove Lemma 4,using the result ofTheorem 1. Inother words, assume (3.1) and (3.2) hold one can show that if 2pots fo)=$f|8sin(na)K,(0)F(a)ds then ne ra=[FOnow, Jol<¥ the socalled generalized Kontrovich-Lebedev transform. Cosequently one can say that there isanequivalence between the generalized Nehler-Fock and generalized Kontrovich- Aslightly different form ofthe Mehler-Fock transformation and its inversion, can easily beestablished, bymaking use ofthe pair ofKontrovich-Lebedey transforms (3, p.75(76)1, 1pete 70-2[awacwae Gay and oe ott)=i£2rea Gay That As, Theorem 3. Te 1yore fa=BPsoryends,Cth) o. thenote g(a)=oi$0)TE,Wd,—Revcd, —dol<¥ G14) where PHandQHaretheassociated Legendre funetions offirst andsecond kindrespec- tively. steernaetvelys ye popia po0 [a ge(hy,Oia The proof of the above theoren ison the sane lines as the proof of Theoren 1, the Linear operator used in the proof will now be 180 c.mast airesyar=[eyArenay Which onapplying tothe equation (3.13), vill reduce 1ttothe form (3.11). Then lasing the inversion (3.10), will produce equation (3.16), asdesired. REFERENCES 1. ERDELYI, A. etal. Higher Transcendental Functions Vol. I,Batenan Manuseript Project, MeGcav-Hill Book Cov, New York, 1953. 2. SNEDDON, IAN. The Use ofIntegral Transforms, MeGraw-Hill Book Cox, New York, 1372 3. LOWNDES, J.S. Note onGeneralized Mehler-Pock Transform, Proc. Camb. Philos. Soc. 16,(1964), 57-59, 4, ZBUANIAN, A. The Kontorovich-Lebedev Transformation on Distributions of compact Support and Ite Inversion, Math. Proc, Caabridge Philos. Soc: 77(1975)> its, 5. ERDELYE, A. etal, Higher Transcendental Functions Vol. 11, Bateman Manuscript Project, MeGrawHill Cons New Yorks 1953. 6, ERDELYE, A, Tables ofIntegra! Transforms Vol. 1,Bateman Manuscript Profect, wecrawHiIL Book Co. New York, 1953,