Weber Schafheitlin-type integrals with exponent 1

Transcription

1 Integral Tranform and Special Function Vol., No., February 9, Weber Schafheitlin-type integral with exponent Johanne Kellendonk* and Serge Richard Univerité de Lyon, Univerité Lyon, Intitut Camille Jordan, CNRS UMR 58, Villeurbanne Cedex, France (Received 7 April 8 Explicit formulae for Weber Schafheitlin-type integral with exponent are derived. The reult of thee integral are ditribution on R +. Keyword: Weber Schafheitlin integral; Beel function; hypergeometric function. Introduction Let J μ denote the Beel function of the firt kind and of order μ. An integral of the form κ ρ J μ (κ J ν (κdκ ( for uitable μ, ν, ρ R and R + = (, i called a Weber Schafheitlin integral [, Chapter 3.4]. If ρ i trictly le than, the reult of thi integration i known and can be found in many textbook or handbook; ee for example [3,,]. However, the critical cae ρ = turn out to be of coniderable interet in the cattering theory (forthcoming paper, The Aharonov Bohm wave operator reviited, by J. Kellendonk and S. Richard. Therefore, we provide in thi paper the reult of Equation ( for ρ = a well a the reult of the related integral where H ( μ κh ( μ (κ J ν(κ dκ ( i the Hankel function of the firt kind and of order μ. We emphaize that both reult are not function of the variable but ditribution on R +. We alo mention that in [] the pecial cae ν =±μ of Equation ( and ν = μ of Equation ( have already been explicitly calculated. It i intereting to note that for value of ρ trictly maller than, integral ( can alo be een a a very pecial cae of a more general family of expreion analyed by Srivatava, Miller and *Correponding author. kellendonk@math.univ-lyon.fr ISSN print/issn online 9 Taylor & Franci DOI:.8/

2 48 J. Kellendonk and S. Richard their collaborator. For example, in [9] the integral n t ϱ J νj (x j t dt j= are evaluated in term of Lauricella hypergeometric function. In [4 7], the Mellin tranform of the product of two generalized hypergeometric function i invetigated, and again Equation ( can be obtained by a very pecial choice of the parameter. However, the extreme value correponding to ρ = are not conidered in thee reference. Such an invetigation would certainly be of interet, and our reult can be een a a firt tep in thi direction (ee alo [8] for related work.. The derivation of integral ( Let u tart by recalling that for z C atifying π/ < arg(z π one ha [, Equation 9.6.4] H μ ( (z = iπ e iπμ/ K μ ( iz, where K μ i the modified Beel function of the econd kind and of order μ. Moreover, for R(z > and ν + > μ the following reult hold [, Section 3.45]: κk μ (zκj ν (κ dκ = z ν + + ; ν + ; z, where F i the Gau hypergeometric function [, Chapter 5]. Thu, by etting z = + iε with R + and ε> one obtain I μ,ν ( + iε := = iπ κh μ ( (( + iεκj ν(κdκ e iπμ/ ( i + ε ν + + ; ν + ; ( + iε. Taking into account [, Equality 5.3.3] one can iolate from the F -function a factor that i ingular for = when ε goe to : F = + ( + iε F Furthermore, by inerting the equalitie + ; ν + ; ( + iε ( + iε = ( + iε ; ν + ; ( + iε. (( + ε / iε

3 Integral Tranform and Special Function 49 and ( i + ε ν = e iπν/ ( + iε ν, in thee expreion, one finally obtain that I μ,ν ( + iε i equal to iπ eiπ(ν μ/ F ( + iε ν (( + ε / iε ; ν + ; ( + iε. (3 We are now ready to tudy the ε-behaviour of each of the above term. For the particular choice of the three parameter (ν + μ/, (ν μ/ and ν +, the map z F ; ν + ; z, which i holomorphic in the cut complex plane C \[,, extend continuouly to [,. The limit from above and below yield generally two different continuou function and, by convention, the hypergeometric function on [, i the limit obtained from below. Since I ( ( + iε <, the F -factor in Equation (3 converge to F ((ν + μ/,(ν μ/; ν + ; a ε, uniformly in on any compact ubet of R +. For the other factor, let u oberve that ( + iε ν converge to ν a ε uniformly in on any compact ubet of R +. Furthermore, it i known that lim ε ( (( + ε / iε = Pv / + i π δ(, (4 where the convergence ha to be undertood in the ene of ditribution on R +. In the lat expreion, δ i the Dirac meaure centred at and Pv denote the principal value integral. By collecting all thee reult one can prove the following propoition. PROPOSITION For any μ, ν R atifying ν + > μ and R + one ha κh μ ( (κj ν(κdκ = e iπ(ν μ/ δ( + ( iπ eiπ(ν μ/ Pv / ν ; ν + ; (5 a an equality between two ditribution on R +. Remark A priori, the econd term in the r.h.. i not well defined, ince it i the product of the ditribution Pv( with a function which i not mooth, or at leat differentiable at / =. However, by uing the development of the hypergeometric function in a neighbourhood of

4 5 J. Kellendonk and S. Richard [, Equation 5.3.] it i eaily oberved that ( Ɣ((ν + μ/ + Ɣ((ν μ/ + ν + μ F ; ν + ; = + ( h( (6 with a function h that belong to L loc (R +. Thu, for any α R, the econd term in the r.h.. of Equation (5 i equal to [ ( iπ eiπ(ν μ/ α Pv / ( + α ν α / ] ; ν + ;, (7 with the econd term in L loc (R +. Clearly, thi ditribution i now well defined. The parameter α ha been added becaue it may be ueful in certain application. Remark To decribe the ingularity at =, the decompoition (7 of the econd term of the r.h.. of Equation (5 i certainly valuable. However, it eem to u that thi decompoition i le ueful if one need to control the behaviour at =. Remark 3 In the pecial cae μ =±ν, the hypergeometric function i equal to, and thu the r.h.. implifie dratically. Proof of Propoition (a For any ε>, let u define the function p ε : R + (( + ε / iε C and the function R + q ε ( C by q ε ( := iπ eiπ(ν μ/ ( + iε ν Ɣ((ν + μ/ + Ɣ((ν μ/ + ; ν + ; ( + iε. Clearly, one get from Equation (3 that p ε (q ε ( = I μ,ν ( + iε and from the above remark that lim q ε( = ν eiπ(ν μ/ ε iπ ; ν + ; =: q (, the convergence being uniform in on any compact ubet of R +. Furthermore, it follow from [, Equation 5..] that q ( = (/iπe iπ(ν μ/. (b Let g Cc (R +, i.e. g i a mooth function with compact upport in R +, and et M g := up q ε (g( d. ε [,] R + For any η>, one can then chooe a compact ubet K η of [, uch that up ε [,] up p ε ( R + \K η η 6M g, which implie that R + \K η q ε (p ε (g( d η/6 for all ε, ε [, ].

5 Integral Tranform and Special Function 5 (c Finally, one ha q ε (p ε (g( d = q (p ε (g( d + (q ε ( q (p ε (g( d R + R + R + = p ε (q (g( d + R + (q ε ( q (p ε (g( d K η + (q ε ( q (p ε (g( d. R + \K η For ε, the firt term on the r.h.. converge to e iπ(ν μ/ g( + ( ν iπ eiπ(ν μ/ Pv R + / ; ν + ; g( d. Indeed, the convergence of Equation (4 hold in the ene of ditribution not only on mooth function on R + with compact upport, but alo on the product q g of the non-mooth function q with the mooth function g. Thi can eaily be obtained by uing the development given in Equation (6. Furthermore, one ha (q ε ( q (p ε (g( d K η up q ε ( q ( p ε (g( d, K η upp g K η which i le than η/3 for ε mall enough ince q ε q converge uniformly to on any compact ubet of R +. And finally, from the choice of K η one ha (q ε ( q (p ε (g( d R + \K η η/3. Since η i arbitrary, one ha thu obtained that the map p ε (q ε ( converge in the ene of ditribution on R + to the ditribution given by the r.h.. term of the tatement of he propoition. Remark 4 In the previou proof, the Lebegue meaure on R + ha been ued for the evaluation of the ditribution on a mooth function with compact upport in R +. Let u notice that the ame reult hold if the meaure d/ i choen intead of the Lebegue meaure. 3. The derivation of integral ( In the next propoition, the function H μ ( of the previou tatement i replaced by the Beel function J μ. Since H μ ( = J μ + iy μ with J μ and Y μ real on R +, taking the real part of both ide of Equation (5 would lead to the reult. However, ince the real and the imaginary part of the Gau hypergeometric function are not very explicit, we prefer to ketch an independent proof.

6 5 J. Kellendonk and S. Richard For any μ, ν R atifying ν + > μ and μ + > ν, and R + one ha κj μ (κj ν (κ dκ = co(π(ν μ/δ( + ( π in(π(ν μ/pv / μ ( Ɣ((μ + ν/ + Ɣ((μ ν/ + μ + ν F, μ ν ; μ + ; if, Ɣ(μ + ν ( ν + μ F ; ν + ; if >, a an equality between two ditribution on R +. PROPOSITION Remark 5 We refer to Remark for a dicuion on the fact that the ditribution correponding to the econd term on the r.h.. i well defined. Proof of Propoition (a Even if the proof i very imilar to the previou one, two additional obervation have to be taken into account: ( the map z F (α, β; γ ; z i real when z i retricted to the interval [, and ( there exit a imple relation between R(I μ,ν and R(I ν,μ. More preciely, for any R + \{} one ha R(I μ,ν ( = κj μ (κj ν (κ dκ = κj μ (κj ν ( κ dκ = R(I ν,μ (. Thu the main trick of the proof i to ue the previou expreion for (, ince the contribution of the F -function in Equation (5 i then real, and to obtain imilar formulae below for (, ]. However, ome care ha to be taken becaue of the Dirac meaure at and of the principal value integral alo centred at. So, for any (, ] and ε>, let u et z := iε. Since the condition π/ < arg(z π, R( iz >and μ + > ν hold, one can obtain the analogue of Equation (3: κh z ν ( (z κj μ (κ dκ = iπ e iπ(ν μ/ (( + ε / + iε m ε(, with m ε : (, ] C, the function defined by m ε ( := ( iεμ ( μ + ν Ɣ((μ + ν/ + Ɣ((μ ν/ + Ɣ(μ +, μ ν ; μ + ; ( iε. (8 (b We now collect both function, for and >. For that purpoe, let l ε : R + C be the function defined for (, ] by l ε ( = iπ e iπ(ν μ/ (( + ε / + iε and for (, by l ε ( = iπ eiπ(ν μ/ (( + ε / iε.

7 Integral Tranform and Special Function 53 By electing only the real part of l ε, one obtain that R(l ε converge in the ene of ditribution on R + a ε to the ditribution: co(π(ν μ/δ( + ( π in(π(ν μ/pv. / Furthermore, let m ε : R + C be the function defined for (, ] by (8 and for (, by ( + iε ν F ; ν + ; ( + iε. A ε goe to, thi function converge uniformly on any compact ubet of R + to a continuou real function m. Furthermore, thi function take the value for =. Indeed, thee propertie of m follow from the fact that the F -function are real on the [, ], that the convergence from above or below thi interval give the ame value, and that the normalization factor have been uitably choen. The function m i the one given after the curly bracket in the tatement of the propoition. (c The remaining part of the proof can now be mimicked from part (c of the proof of the previou propoition. Therefore, we imply refer to thi paragraph and omit it from the preent proof. Reference [] M. Abramowitz and I. Stegun, Handbook of Mathematical Function with Formula, Graph, and Mathematical Table, National Bureau of Standard Applied Mathematic Serie, Vol. 55, US Government Printing Office, Wahington DC, 964. [] L. D abrowki and P. Šťovíček, Aharonov-Bohm effect with δ-type interaction, J. Math. Phy. 39( (998, pp [3] I.S. Gradhteyn and I.M. Ryzhik, Table of Integral, Serie, and Product, Academic Pre, New York, 965. [4] A.R. Miller, On the Mellin tranform of product of Beel and generalized hypergeometric function, J. Comput. Appl. Math. 85( (997, pp [5] A.R. Miller, On the critical cae of the Weber Schafheitlin integral and a certain generalization, J. Comput. Appl. Math. 8(, (, pp [6] A.R. Miller, The Mellin tranform of a product of two hypergeometric function, J. Comput. Appl. Math. 37( (, pp [7] A.R. Miller and H.M. Srivatava, On the Mellin tranform of a product of hypergeometric function, J. Autral. Math. Soc. Ser. B 4( (998, pp. 37. [8] R.N. Mirohin, An aymptotic erie for the Weber Schafheitlin integral, Math. Note 7(5, 6 (, pp [9] H.M. Srivatava and H. Exton, A generalization of the Weber Schafheitlin integral, J. Reine Angew. Math. 39 (979, pp. 6. [] G.N. Waton, A Treatie on the Theory of Beel Function, nd ed., Cambridge Univerity Pre, Cambridge, UK, 966. [] A.D. Wheelon, Table of Summable Serie and Integral Involving Beel Function, Holden-Day, San Francico, 968.