Two integrals involving products of two Bessel and a generalized hypergeometric function

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

ELSEVIER

Journal of Computational and Applied Mathematics 59 (1995) 381-384

Letter Section

Two integrals involving products of two Bessel and a generalized hypergeometric function H a r r y A. M a v r o m a t i s t Physics Department, King Fahd University of Petroleum and Minerals, P.O. Box 58, Dhahran 31261, Saudi Arabia Received 10 January 1994; revised 25 February 1995

Abstract In this paper two integrals involving products of two Bessel and a generalized hypergeometric function are obtained. Special cases of these integrals are useful in inverse scattering theory. An error in a standard literature integral listing is also pointed out. Keywords: Bessel; Generalized hypergeometric; Inverse functions

In the literature [1, p. 153], [2, p. 854], [7], definite integrals, involving the products of two Bessel and various other functions, are given. The author recently obtained two integrals, involving the products of the square of a Bessel function and generalized hypergeometric functions, which are useful in the context of the inverse scattering problem [3]. O n e of these results is generalized in the present paper, and an additional similar integral is also derived. The mathematical results of this paper can be displayed succinctly as two theorems: Theorem 1. 3F2(2,2(o

"

+v),½(a--v);1

/~,1 +/./; -4to2)codco

#p~- 2 K~(p) = - 2 ~ - l F ( ½ ( a +v))F(½(a - v ) ) (Rea>l+lRevl,#=l+½,1aN,

Re½(cr+v)>0,

1KFUPM support is acknowledged. 0377-0427/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 7 7 - 0 4 2 7 ( 9 5 ) 0 0 0 3 5 - 6

Rep>0).

H.A. Mavromatis / Journal o f Computational and Applied Mathematics 59 (1995) 381-384

382

Theorem2. foJ,, (ow) J,, + 1 (op) 3 F2 (~,½ (~ +v),½ (~ - v ) ; 1 - #, 2 + #; -4o92)co 2 d o #(It + 1)p~-3K~(p) = - 2 ~- ~r(½(~ +v))r(½(~

-v))

(Rea>2+lRevl, l l = l + ½ , 1 e ~ ,

Re½(a+v)>0,

Rep>0).

Theorem 1 may be obtained by starting from the following integral I-1,2]:

f f x ~- 1 pFq(al,...,ap; bl .... ,bq;

2x 2) Kv(xy) dx

=2~-2y-'F(½(tr+v))F(½(a-v,)p+2F,(al,...,ap,½(tr+v,,½(a-v);bl

.... ,b,; - 4~), (1)

where Rey > 0, p -% ]Revl, Kv is related to the Hankel function [2, pp. 951, 952] n~l)(z) = J~(z) + iNv(z), zi en/2 vi H~l) (iz) K,(z) = -~ and the pFq is a generalized hypergeometric series [2, p. 1045]:

nFm(al,...,an;bl, "'" ,bin;x)= ~ (~)k--.-(b-~kkk! (al)k...(a,)k xk' k=O

(2)

where the Pochhammer's symbol (C)k:= F(c + k)/F(c), the series terminates if any of al .... , a, are negative integers, and is in general convergent if lxl < 1. The listings of expression (1) are in error insofar as they list the 42/y 2 term in the generalized hyl~rgeometric function p+2Fq without a minus sign. Eq. (1) can be proved for arbitrary v by expanding pF~ using the series expansion Eq. (2) (which is allowed because p ~< q - 1) and using [2, Formula 6.561.16, p. 684], which in turn can easily be proved. With the identifications p = 1, q = 2, 2 ~ 22, a~ = ~2, b~ = ½ - l, b2 = l + ~, 2 Eq. (1) reduces to: fo[

,2-1,1+~; rc4x22Z (l + ½) 1F2 ( 23..1.

=

2"-~y-'r(½(a

-22x2) ][ -

+ v))r(½(G - v))3F2

(

x'-3(14~K~(xy)rc] dx

~, ½(~ + v),½(~ - v);½ - l, l + a2;

= fogu(x2)[ _ x°-3(l +T4~½)K'(xY)Tr-]~ Jox,

4)'2"~ ~-~ ]

(3)

where [5] the function: 4x222 g,l(X,Z) -

~(l + ½) 1F2(~;½ - l, l + ~; - x2,Z ~)

(4)

H.A. Mavromatis/ Journal of Computational and Applied Mathematics 59 (1995) 381-384

383

is the inverse [4] of j2 (x'2):

foj2 (x'2)gu(x2)

(5)

d2 = ~-;~ {5(x' - x) + 6(x + x') - ( - 1)t26(x)}.

Multiplying Eq. (3) by

22j2(x,2 ) = 2n 2 ~--x,J,+ ~/2(x'2), and integrating over 2, where x, x' > 0, and Re a > 1 + IRe v[ (so that the origin), one obtains finally:

fo

J~(ogp)3F2(~, ½(a +v),½(a - v ) ;

[x'-XK~(xy)]

vanishes at

#P'r-2Kv(P) 1 - # , 1 + # ; -4o92)o9do9 = - 2~_XF(½(a +v))F(½(a -v))

( R e a > 1 + IRevl).

(6)

If v = _+ ½ this reduces to a result derived previously [3]. T o obtain T h e o r e m 2, one uses similar arguments, and the function [5]:

gl l +

1 (X2) =

--

which is the inverse T h e o r e m 2:

2)3 n(l +4 ( X½)(l + ~2)1F2(~2;

ofjl(x'2)j~+l(x'2).

- l + ½,1 + ~2; - (x2)2),

Starting from Eq. (1) one obtains for R e a > 2 + IRe vl,

f)J,(ogp)J~,+, (cop)3F2 (~, ½(a +v),½(a -v); #(# + 1)p~'-3K~(p) +v))F(½(a --v))

= -- 2 " - l F ( ½ ( a

(7)

1

2 + # ; -4¢o2)~o 2 de)

( R e a > 2 + IRevl).

(8)

Finally one notes that since expressions (4) and (7) are valid inverses for integral values of l, one has the additional constraint in Theorems 1 and 2 that # = l + ½, l ~ N, while in order to avoid singularities of the G a m m a functions one additionally requires Re ½(a + v) > 0. An application of these integrals in inverse scattering theory arises if one wishes to obtain certain weighted integrals of an u n k n o w n scattering potential V,(r) which produces the scattering phase shift 6,(k) when a particle of mass M impinges on it with an energy E = h2k2/2M. In particular suppose one wishes to obtain

f o V,,(r)r~- ~K~(r)dr = f ) V.(r)r"- 2r K~(r)dr,

(9)

under the assumption that

M fo

tan ~,(k) = - ~

V~(r)J~

(kr) r dr,

(10)

i.e., assuming the partial-wave Born approximation [6, p. 244] is valid for this scattering process.

384

H.A. Mavromatis / Journal of Computational and Applied Mathematics 59 (1995) 381-384

Using Eq. (6) one has

f o V~(r)r~- ' Kv(r)dr = f o V~(r)r~'-2Kv(r)r dr )

=

x aF2(a2,½(# +v),½(# --v); 1 - o ' , 1 +or; - 4 k E ) k

dkrdr,

Vo(r)J~(kr)rdr

= _ 2"-~F(½(~ + v))F(½(# - v)) G

x aF2(~,½(~ +v),½(~ - v ) ; 1 - e , 1 + e ;

2.-lr(½(

+

-

~Mrc

v))h ~

°

Jo

-4k2)kdk,

tan6o(k)

-4k2)kdk, the 6o(k) are known.

x 3F2(~2,½(/2 +v),½(# --v); 1 --o-, 1 +or; where the integral on the right-hand side can be evaluated since If, for example,/~ = 2t7 + v + 2, Eq. (11) simplifies to:

(11)

o V~(r)r2~+v+1K~(r) dr =

22°+V+1F(o " + 1)F(tr + v + 1)h 2 ~,oo trMrc J o tan6~(k)2F~(a2't7 + v + 1;1 - t7; - 4 k 2 ) k d k .

(12)

References [1] Erd61yi et al., Tables of Integral Transforms, Vol. 2 (McGraw-Hill, New York, 1954). [2] I.S. Gradshteyn and I.M. Ryzhik, Tables of Integrals, Series and Products, Fourth (English) edition, prepared by A. Jeffrey (Academic Press, New York, 1980). [3] H.A. Mavromatis, Two integrals arising in inverse scattering theory, J. Math. Anal. Appl. 188 (1994) 458-464. [4] H.A. Mavromatis and A.M. A1-Jalal, On obtaining the scattering potential and its moments in the partial-wave Born approximation, J. Math. Phys. 31 (1990) 1181-1188. [5] H.A. Mavromatis and K. Schilcher, Inverse functions of the products of two Bessel functions and applications to potential scattering, J. Math. Phys. (1968) 1627-1632. [6] E. Merzbacher, Quantum Mechanics (Wiley, New York, 2nd ed., 1970). [7] A.P. Prudnikov, Y.A. Brychkov and O.I. Marichev, Integrals and Series (Gordon and Breach, London, 1986).