THE LAPLACE TRANSFORM OF THE MODIFIED BESSEL FUNCTION K(t±»>x) WHERE m-1, 2, 3, ..., n

by F. M. RAGAB (Received in revised form 13th August 1963)

1. Introduction In the present paper we determine the Laplace transforms of the modified ±m Bessel function of the second kind Kn(t x), where m is any positive integer. The Laplace transforms are given in (2) and (4) below, p being the transform parameter and having positive real part. The formulae to be established are as follows (l)-(4): f Jo 2m-1 2\-~ -i-,lx

2m 2m 2m J k + v - 2 2m (2m)2mx2 v+1 v + 2 v + 2m (1) 2m 2m 2m where R(k±nm)>0 and x is taken for simplicity to be real and positive" When m = 1, x may be taken to be complex with real part greater than 1. The asterisk in the generalised hypergeometric function denotes that the . 2m . ., v+1 v + 2 v + 2m . factor =— m th e parametert s , , ..., is omitted; m is a positive 2m 2m 2m 2m integer.

m m 1 e~"'Kn(t x)dt = 2 ~ V f V J ±l - 2m-l 2 2m

2m 2m 2m J 2m v+1 n v+1. 4p ' 2 2m 2 1m"' (2m)2mx2 2r2m+l v+1 v+2 v + 2m (2) _ 2m 2m ~lm~ E.M.S.—Y

Downloaded from https://www.cambridge.org/core. 02 Oct 2021 at 05:34:47, subject to the Cambridge Core terms of use. 326 F. M. RAGAB where m is a positive integer, R(± mn +1)>0 and R(p) > 0; x is taken to be rea and positive and the asterisk has the same meaning as before.

l n n k k + i einx2 V v(i - k + 2m-l .(3) i,-i i \ 2 2 2m 2m 2m "M2m)2ml' where R(x)>0, R(k)>0 and m is any positive integer. f Z\ dt =

2m-l 1 •(4) i.^i 7 2* 2'2m'2'2'-m ' 2m where R(p)>0 m is any positive integer and JC is real and positive. In (3) and (4) 5] means that in the expression following it i is to be replaced by — i and the two expressions added. The function appearing on the right of (3) and (4) is MacRobert's ^-function whose definitions and properties are to be found in ((1), pp. 348-358). These formulae will be proved in section 3 by means of a subsidiary formula which will be proved in section 2. The following formulae are required in the proofs ((2), p. 77):

; a/- qi Ps: = n cosec Jo z , cc2, ..., ctp; e

k_ k+1 k+2m-l " 2~in' 2m 2m k + v^ cosec | | n 2m k + v : e±2im\2m)2mz ~2m' 2m v+1 v + 2m k+v K (5) 2m' 2m , 1 + 2m 2m

where p^.q + 1, m is a positive integer, i?(mar + A:)>0, r = 1, 2, 3, ...,/>, | amp z \

Downloaded from https://www.cambridge.org/core. 02 Oct 2021 at 05:34:47, subject to the Cambridge Core terms of use. THE MODIFIED BESSEL FUNCTION K(t±mx) 327 holds if the integral is convergent. ((1), p. 406):

2m f'V'W-'ECp; a,: q; ps: z/X )dX Jo Z i; «,: q; ps- „ ^A (6)

where m is any positive integer, R(k)>0, a.+v = , v = 1, 2, 3, ..., 2m. 2m ((l),p. 352): 1 E(p; ar: q; ps: z) = £ ft r(as-ar){n r^-a,)}- ]^,) r = 1 s = 1 p ^ „ r«r,ar-p1 + l)...,ar-ap + l: (-l) -«z-| Z 4+l^p-l > V'J L^-aj + l, ar-a2 + l, ...*..., ar-ap+lj where p^^ + 1 and r = 1, 2, ..., p. The prime in the product sign signifies that the factor for which s is equal to r is omitted.

where p^q r(z)F(l-z) = n cosec nz (9)

2. The subsidiary formula The formula to be proved is

(10) 2 2 4

where Kn{x) is the modified Bessel function of the second kind and £ has •', -> the same meaning as in (4). To prove (10) expand each E function by means of (7) and combine the two resulting expressions by omitting common terms; then applying (9) the right side of (10) becomes

2 sin nn x X -{r(l + «)}" \ A 0Fi I ; l + n; ^-) = ^n(x), \2J \ 4 / J and the formula is proved. 3. Proofs . , e'*x2 In (5) take q = 0, p = 3 with o^ =1, a2 = £«, a, = -+«, write

Downloaded from https://www.cambridge.org/core. 02 Oct 2021 at 05:34:47, subject to the Cambridge Core terms of use. 328 F. M. RAGAB

for z, then for z, apply (5) twice making use of (10) and get

2 -, k-x y l ( n fi.. "t " m t E 1 e = 7i cosec kn(2n) ~ f 1 AiT2' 2" 4 Jo

- A: . k + 2m-l 1- —, 1- , ..., 1 2m 2m 2m

sin I I n \2m n . A:+v « , A:+v. , ,, - ft2 1 + : eim2'm\2mf 2m 2m 2 2m 2m v+1 v + 2m — , ...*..., — 2m 2m 2m where m = 1, 2, 3 Now change each JE'-fun.ction to a generalised hyper- geometric function by means of (8), noting that the first two series cancel, apply (9) and (10) and so obtain (1). (2) can be deduced from (1) by writing pt for t and taking k = 1, then writing — for x. & pm e±inx2 Proofs of (3) and (4): To prove (3), apply formula (6) with for z,

taking p = 3, q = 0 with a.x = 1, a2 = £n, a2 = — \n. Combine the two results using (10) and so obtain (3). Formula (4) can easily be deduced from (3) by taking k = 1 and writing xpm for x. It may be noted that the first two series which are obtained by expanding any of the two E functions appearing on the right of (4) by means of (7) cease to exist because ax = a2 = 1. These two series are replaced (see (3), p. 30) by J where

Ai = ^(A)-logz + \

Downloaded from https://www.cambridge.org/core. 02 Oct 2021 at 05:34:47, subject to the Cambridge Core terms of use. THE MODIFIED BESSEL FUNCTION K(t±mx) 329

Here #(z) = ^ log F(2 +1) dz and z = 4(2m)2m'

REFERENCES (1) T. M. MACROBERT, Functions of a complex variable (London, 4th ed., 1954). (2) F. M. RAGAB, Integrals involving ^-functions, Proc. Glasg. Math. Assoc, 2 (1953), 77. (3) T. M. MACROBERT, Evaluation of the .E-function when two of the upper parameters differ by an integer, Proc. Glasg. Math. Assoc, 5 (1961), 30-34.

MATHEMATISCHES INSTITUT DER UNIVERSITAT BONN, GERMANY AND FACULTY OF SCIENCE CAIRO UNIVERSITY CAIRO, U.A.R.

Downloaded from https://www.cambridge.org/core. 02 Oct 2021 at 05:34:47, subject to the Cambridge Core terms of use.