The Laplace T,'ansform of the Bessel Functions where n ~ 1, 2, 3~ ... by F. M. RAhaB (a Cairo, Egitto - U.A.R.) (x) oo Summary. - The integrals / e--PtI(~ (1)xt~-n tl~--~dt and f e--pttk--iJ f 2a~t=~ni 1] di, n=l, 2, 3, .. o o are evaluated in term of MacRoberts' E--Functious and iu terms of generalized hyper. geometric functions. By talcing k~l, the Laplace transform of the four functions in the title is obtained. § 1. Introduction. - Nothing is known in the literature ([1] and [2]) of the Laplace transform of the functions where n is any posit.ire integer, except the case where the argument of the Bessel functions is equal to [~cti/2] (see [1] p. 185 and p. 199). In this paper the following four formulae, which give the Laplace transform of the above functions, will be established: / co 1 (1) e-Pttk--iKp xt dt=p--k2~--~(2u)~-'~x ~ o ](2n)2"p2]~_k nk ~ h n; : e :ei~ [~4x-~--]2 E A n; ~ + 2 , 2 2: (2n)~"p ~ 1 [[27~2n~2~ i . 1 E ~ n; 2 ~- , A n; 2 : 2: (2n)2"p 2 e:~ where R(k + U) ~ o, R{pI ~ o and n is any positive integer. The symbol h(n; a) represents the set of 318 F.M. RAGAB: The Laptace Tra.ns]orm o~ the Bessel Functions, etc. parameters o~ a+l a+n--1 8 (2~ e-ptt~-I xt dt = 2 k-~ r: o [ 1 1 1 ( 1 ) ( ~, 1 I, k, k+ h n; ~ h n ~, -~i ~ 2 )' ~ , ; --~. ~ ::4,~+:n2. e~ where R(k) > o, R(p) > o and n is any positive integer. The symbol A(n;~) has the same meaning as in (1) and the symbol Z means that to the expression following it a similar expression will --i instead of i is to be added ; oO 2V ~ 2Y 1 (3) 0 X=o k! l~(2v --{- ~, + 1) [n'nP" l k v+). k+l v+~. ~i.= 4a" , -- " e n2op 2 2+ n 2 + n ' -- ~F2.-: X+l k+n 5(n; 2v+l) where and n is any positive integer. The symbol A h~s the same meaning and • t . the asterisk denotes that the factor -m the set of parameters n X+I ),+2 X+n n n is omitted; I] v k --v---1 dt --p --k+-9,.~ 0 F. M. RAGAB: The Laplace Transform of the Bessel Fu, nctio~s, etc. 319 I p~ l~_~_ ~ +~k oF2,~+~ ; A n; 1+ --v A; n; 1-~ nk + v 2 ' 2 ' 2 ; 4(nx) 2ne±~n~ nk -t- n -k v] 3 p~e -~'~] oF~.+l[; ~(n;1 + .k +~2 -~)' A(,~; 1 + 2 ] '2;~J Y --V -}- p-~+ ~ m (__ 1)z2,Tn~z n I p2 iz_ z! v(v +)` + 1) ~" Y, ),+1 ),+n 1--k }_v-~2), 2--k v-~2)` ' oF2~+1 ; n n ' ~ - , ~ + - ,I "Pze~=inrc ] _ A(n; 1 + v + )`) : 4(nx)~ . where R (k--~ v) >o, R(p)>o and x is taken for simplicity to be real and positive. The function appearing on the right hand sides of (1)and (2) is MAcRoBER~'S. E function whose definitions and properties are to be found in (3} pp. 348-358. The symbol A and the asterisk have the previous meanings. These four formulae will be proved in § 3 by means of sub- sidiary theorems which will be stated an proved in § 2. In § 4 some known results and other new results will be derived as particular cases. The following formulae will be required in the proofs. ([4] p. 92, form. (7).) : (5~ f e-)')`~-lE ; ~,, : q ; ~ : ~ o) d), ----- 2 k-17:- 1 0 320 F. 5{. RAGAB: The Laplace Transform of the Bessel F,t~nctions, etc. 1 1 ~ p p i i i ] ~q (2~) ~-~" <~+P-q)+"-~ '~ ~- ~ P~÷ -~P-~ n-- i l • E (-- 1)x[4z-~n-"(~+q-p)]; k ~ k-lrl t 1 ,, p; h(n; a,.+X), 2+n 2 +n:4 z n"(~+q-V)e:~('+~)~ E ~.+--, 1 ......., --,X+n q; A(n; p~+~) n Tt where R(k)> o, p ~> q + 1 and n is any positive integer. For other values of p and q the formula holds provided that the integral is convergent. ([3], p. 352): if p--~q, then (6) E(p; a,.:q; p,:z)= F(p~)r(~)...]:(av) ... F(p~)~F" (p ; a,.: q; ~; _~zl) . ([3], p. 353): if p_>.q+l, then E(p ; ~ : q ; p~ : z) = Z; IIP'P (a, -- a,. )l lIv r(~t-- a,.) I-~ r(~Az% F (a,., a,.-p,+l, ..., a,.-- gq -}- l; (--1)p-qz) (7) \a,. -- a~ + l, ... ~ , a,.--a~+l ([3], p. 409): if p~q+l, then p E(p; a,.: q; p~ l z)-~--uP-q-lE [I sin (pt--a,.)u a,., ~ -- ;~ + ;q + 1: e~:~:(v-q-1)z t (8) I 1~' sin (a s-- ~,-)r:l-lE ( :or--al+ 1, ... * ..., a,,-- a~+l ([3], p. 411): (9) E(: v+l: z)=z~J~ (13), p. 154, ey. 5): F. ~. RAGAB: The Laplace Tra~tsform o] the Bessel F~a~wtion.s~ etc. 321 n-- 1 1 1 1 Also the followi.ng formulae are required: (11) r(z)t'(t -- z)= r~/sin ~z; k~ (k + 1)~ (k + n -- 1)7: (12) sin -- sin ... sin --- 21-n sin kr~ Tt ~ n (13) sin sin -- ... sin -- -- 21-'n Tt n ~ For the derivation of the particular cases, use will be made of the following formulae : F 2, =Vv i+ Tvc ) (14) can be obtained by differentiating the expansion 1 {l+Vl--zt--2b (15) F(b, b+ ~ ; 2b + 1; z) "-- t 2 t which can be extablished by means of LAC~nA~GE expansion, and then 1 putting b=~--l. ([5], p. 103, ex. (38)): (16) c(1--z)F(a, b; c; z) -- oF(a --1, b; c; z) ------(o--b)~F(a, b; c+ l; z) ([3]; p. 305); (17) P~'-v(z) "-- F(v + 1) ze-v~- ' 2 ' where P,-~(z) is the associated LEG]~I,~Dm~ finetion of the first kind. And the known transformation (18) F(a, b ; c; z) -- (1 -- z)C-'~-~'F(c -- a, o -- b; c; z) Annali dl Matematica 41 322 F. ~. RAGAB: The Laplace Tra~sform of the Bessel Fu~wtio~s, etc. § 2. - Subsidiary formulae - The thorems to be used are: ([6]; p. 304): 1 1 1 ~.\ i 9) K~,(~) = U=1 ~,E _~ f1E 1,~, 2g,::4me ) 1 (20) x -~ Z :E(p; ~+5~, ]: q; 9~+},: d~z) -- i, --i 1 E -:E(p; a,,, I:q; ;~: d "~z) i,--i (20) can be proved by expending each E--function on the left by means of (7) and combining the two resulting expressions by factoring out common terms applying (11). Ifp~q+l, lampz[ ~, Rk+-~-/( 2art>o, r:l, 2, 3, ..., p and n is any positive integer, then Oo -(91) e-lXk-~E p; oq~: q; ~,: zX~ dy= o i i p q I T i 1 - ~ ~+--~." + ~ (2.)(~-~')(~+~-°)~2 -~ ~'-~ ~'- ~-~+~. • k,., ~ (~) --/~;+~, v, ~(z) i where 1 (21') ~, p, q (z) ---~ cosec ~z ~k . .E A n; 1 +-~), .~, q; ~ '*k/ and 3 1 Rk +~. p, q (z) is k, .. q(z) F. ~¢[. RAGAB: The Laplace Transform of the Bessel Functio~s, etc. 323 3 i with (k + 1) instead of k and 2 instead of the 2 which appears in the last line of (21'). Also U"~, p, q(z) is given by ~, ~, q(z) = (-- 1) x cosec ~ -- n 7: cosec 2 7:" E ),.+1 k+n 1 k ), l k ), ~'"'*'"'~' 2 2+~-,' ~ + n, q; h(n; ~+k) Io (2[) the symbol h and the asterisk have the previous meanings. For other values of p and g (21) holds provided that the integral is convergent. Iv prove (21) consider the special case with p ~ 1 and q-- 0, a~= :¢, then the left side becomes co Go 2 z ~ e-Xk~:+~--lE a 0 0 2~ 1 ~--1 ~. -- z~F2 ~+ ~--~-~n ~-~ Z (-- 1)~(4z'*); ~.= 0 • * -- : - e=~(~+l) ~ _by 5. n ' 2 ~- n n n 4 z~ Now expand the last E-funtion by means of (8) taking in (8} a~ = = k a ), k+l ---:¢+)' and o~ t+~--no: +~k+t (t--0, 1, 2,..., n--l), ~ + ~ + n' ~ - 2 -~ n then the last expression becomes CG -~X~-IE ~ :: zX ~ d), -- z~2 k +~-1 ~?-n~-i o (22) ~, (-- 1)~(4z')~ A k Ak+~,~'~ x(z) ).------ 0 2_~ 1 3 ¢$ -- 1 ~--I + z~2 k+ ,~ r:~n~-~ ~ v, Bx", t(z). where sin + ~ t e:~i~(n+~)~-" ~ + A ~' ~(z) = It=o n 2 n n -- 1 '* sic(1 1sin -- +~s)l ~ sin , 2 n s=o 324 F.M. R~GAu: The Laplace Truns]orm of the Bessel Fur~ctio~s, etc. k ik .+~ k ~--1 , - : e-~(n+l)ni4Zn 2+ n '2 4 n r-~, ... ~+ n E ~1 k 1 k I ~, l+~,'"*'",nq- ~ 8 3 and A"'~ (z) is the value of A~',~(z) with k -+- 1 instead of k and ~ instead k+l,~. of the ].1 Also B"~,t(z) is given by ... B~", t(~) = (-- 1)t+:' t=o n n 4 sin I t+n ;~ +e~: .--IIsin--n --n ~sin 2 [ ~-]--- k -]- t, c¢ + t--1 +1,...*..., :¢ + t': e~=(,+~).i4z. n n E t k 1 t k t+l t+n J._ l*n 2' 2--n 2' n ' ' n Here apply (11) and (12) to substitute for each sine product in (22). The n~ 8_ asterisks in the three E-functions appearing in Ak;~(z), A;'g:,x(z) and Bz~,t(z) denote that the three parameters k ~.