BOLLETTINO UNIONE MATEMATICA ITALIANA

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1 BOLLETTINO UNIONE MATEMATICA ITALIANA F. M. Ragab Integrals involving products of E-functions, Bessel-functions and generalized hypergeometric functions. Bollettino dell Unione Matematica Italiana, Serie 3, Vol. 2 (957), n.4, p Zanichelli < L utilizzo e la stampa di questo documento digitale è consentito liberamente per motivi di ricerca e studio. Non è consentito l utilizzo dello stesso per motivi commerciali. Tutte le copie di questo documento devono riportare questo avvertimento. Articolo digitalizzato nel quadro del programma bdim (Biblioteca Digitale Italiana di Matematica) SIMAI & UMI

2 Bollettino dell Unione Matematica Italiana, Zanichelli, 957.

3 Intégrais involying products of E-functions, Bessel-functions and generalized hypergeometrie functions. Nota di F. M. EAGAB (al Cairo - Egitto) Sonto. - GH intégrait infiniti menzionati nel titolo sono calcolati per mezzo di funzioni E e di funzioni ipergeometriche generalizzate i cui sviluppi asintotici sono noti. Vengono anche calcolati molti integrali per mezzo di prodotti di funzioni di Bessel Snmmarj', - The infinité intégrais mentioned in the title are evaluated in terms o f E-functions and generalized hypergeometrie functions whose asytnptotic expansions are knowm. Also many intégrais are evaluated in terms o f products of Bessel functions. Introductory : In 2 the following formula will be established. If when p>q-i-l, Z > m -+-, R(x r -+- k) > 0, r =, % 3,..., p, B{2$i k)>0, t =, 2, 3,..., l and amp z \ < w, 00 ƒ ; a r : q ; p s ; : m ; <J U : X ir 2 cosec [^ 2' 2 ' 2 ^w *"' 2^qj 2 ^" + ~ )* "* ' 'i 2*'-' -+- ' re* sec ' 2^ 3 2' c i 3 2 H- ),... * 2^ k ~ 2' gtpi " p + )» 2( a i H - 2),.. ' 2 (^' 2^P«2(Pi + ' 2),..., 2' fch " 2"

4 536 F. M, RAGÀB 2TT 2 cosec XE 2 (h H- 04),..., g [h -+- a ), g (fc 4- a A + ),..., - (fe -H a p ), ^ & 2 (pl For other values of p, q, l, m the resuit holds if the intégral is convergent. In 3, many particular cases of formula () will be considered. The following forraulae will be required in the proof. If p > q -H, B [k + g a r j > 0, r =, 2, 3,..., p, amp z < TT, o oo \ a,.: g; ï.r X *P» 2 ^! "*" ) î *" ' 2 ( (a, > 2 ), (a, + 2),..., g{a p + 2) : (~ k)r(i «- 70»)* g a, -K fc,..., 2«p,... (2).

5 INTEGKALS INVOLVING PRODUCTS OF ^-FUNCTIONS, ECC. 537 Por otter values of p, q, the formula holds if the intégral is convergent, [I]. j E(p; * r :q;?! : z/u)du = E(p + ; a ; q;. Pl : s),... (3) o (see [2] p. 384 ex 06); ; * P : g ; p s : fe)<k = ^(P Ï «,.:g+i; p. : «),». (4) where the contour starts at oo on the Ç axis, passes positively round the origin and returns to oo on the Ç axis, the initial value of amp Ç being TT, (see [2] p. 394 ex 05). 2. Proof of the formula : In formula (2), replace X by l 2 jz 2 and then replace A; by ^k and z by V(«)) so obtaining ""*' 2 pl ' ""' 2 sec (^ x E 2 (a l -f- ),..., g (a p -h ), g (*! H-2),..., g i 3 3 i 2 ' 2 ~ 2 &> 2 (pi "*" )ï " ' 2 (p? + )! 2 (pl 27T* cosec where #>g-+-l, i?(a r H- A;) > 0, r =, 2, 3,..., p and amp z < TT. 35

6 538 F. M. EAGAB This is a particular case of formula (). In fact it is formula () with l m = 0. On replacing z by zju and applying formula (3) repeatedly; and then replacing z by tz and applying formula (4) repeatedly; the gênerai case can be deduoed Applications: We are now in a position to evaluate a large number of infinité intégrais by means of (). These intégrais involves products of BESSEL functions, confluent hypergeometric functions. For the définitions and properties for the i -functions see [2] p In () take p q = lr=m 0, then it becomes ƒ o o exp 5 r (*)»'V( (s*)»'v,(;g, -*;-i/4. because if p,... (5), In () take p=l 2, q = m = O with a,=r^+%, oc 2 = ^ n, i Sj = -= -+- m, [3 2 == g m ; replace X and 0 by 2X and Sz ; then from / \ the formula cos WTCÜM^ -H % ^ n ;* 20) = V(27r^)e 3 Zr )ï (^);,... (7), (see [2] p. 35), it follows that if B(l dt 2m &) > 0, i?(fc-+-5=b^j>0, I amp ^ <TT exp (X -+- 0/X 2 )X te ~"2 jj

7 INTEGEALS INVOLVING PBODUCTS OF i?-functions, ECC. 539 i 7 O 7 2ï (ViO~ l COS»TC COS»W7C , XE j T 2 COS TI-K C08 W7C « o 3 3! 2' 2~2 fc 2~ 2 7t~2^ cos nn cosw^tt s~"2,-3,.,3, i ^ 2v2 + 2 ^ i,... (8). In () take p 2, g = 0, otj = n, ^ == H n fey 2X and 4», then from (7) and the formula î replace X and E U fc' -+- m\ 2 k' m '." r( - t '+- w ')r(a-ft'-m')0- k 'ei-w k%^(0),...(9), i 00 oc exp (X H- ^ /l -\ cos (mt) cosec I Ö ^^ ) i f \ /l s & -*- w ' r ( s A;. m! \^

8 540 F. M. RAGAB xe lhh2i 4 v2 J ft 3H ~~ 2 h2n -3H 4 ' 4 ' & O 2ilfh 4 ', / 2 ^-+-^ 2 cos /l \ W7c) sec t 25/2 r f - fc' H- w'] r (~ m') 3+2n 5-H 2n 3-2n 5-2n,,, 7, #, T~~ s J i 7 ; J ) & -hw ijfc^ fc m ö Jfe T cos (WTÏ) cosec (nk) k, 23/2 r(l AÏ' -h m')t (l W m'\ ~r~ atv H~ u/b O f ^Tfr H Ö/C atv f- ö/c ö u/t H~ â/c 4 0 h' -+- m' >» fc' w' ; z,... (0), where jb(fc-i-»=twj > 0, JR( 2fc' ± 2m' k) > 0, amps <Tt. Again in () take l = m = 0; then from (9); if jr ffc- - «A;'±wt)>0 and os is real and positive 0 / exp ( X - X»/x) X^-ifffc. m,(x)dx = 2-2fc'-2 7ï i 4 f g *' H- «*'j r f g h' w'j sin 2

9 INTEGRAIS INVOLVING PKODUCTS OF Ü7-FTJNCTIONS, ECC. 54 ( 2k' -H 2m'), ( 2k' 2m'), j (3 2k' +- 2m'), 2' (3 _ 2k' - 2m') : f 2-2^-2-* g ( \ ( \., 4 fc_l C2 2 ( - (3 _ 2k' H- 2m'), - (5 2&' -H 2m'), ^ (3 2k' 2m'), ) 3 3 ( 2' 2 ~2 g - fc' +- wa r / - fc' - m') sin i», 3 X E< ' 2 fc ' 2 " " 2 3-2fc' ^-,... (U). In ^) take j9-g = î = m = l, then from (6)^ it follows that if R(k H- a) > 0, i?(2s k) > 0 and x is real and positive o

10 542 F. M. RAGAB X A* V2' ~2 fc ' 2 P r = j X fc f+fc,...(2). In () take'jp = 2, g = and l = m = ; then if.b(fc -t- a,) > 0, +- a 8 ) > 0, 5(2p fe)>0 and 0:=a; where x is real and positive ^, a 8 ; P; - X (p _ &j r ( J fc)

11 INTEGRAL S INVOLVING PRODUCTS OIT EJ-FV NOTIONS, ECC *'2 p '2 p -*-2' ff X (p -H - î * - g ± 3 3 s '.2"2 ' 2^2' 2 p "^ ' 2 + ff " 2 - ) r( - \ fcfcfcfc, ' 2 ' 2 "* 2 ' 2 p H " 2 ' 2 p H ~ 2 2 ',..." (3). 4. Two formulae: The following two formnlae will be established. If when J9>g -f-, i?(a f. -f- k) > 0, r =, 2, 3,..., p, R(p s ) > 0, s, 2,' 3,..., q, amp s < *, JB(fc) > -. oc 2 ( a i -*- & ))» * 2 (**> ~*~ *)» 2 (

12 544 F. M. BAGAB For other values of p, g, the formula holds provided that the intégral is convergent. If when jp^g -+-, B(k H- <x r )> 0, r =, 2, 3,..., p, >O i t = l, 2, 3,..., q, R{k)>2, and h Pi,..., p* / 2' 2 f. s \ 2 2 ' À 7 (oc,h- A; - ),..., n (ot p-t- Aï -), 3 (a à -h &),..., 5 (ot p -+---&) X t/^,... (5). For other values of p and q, the formula holds provided that the intégral is convergent. These two formulae will give some particular cases which will be considered in 5. Proof of the formulae: To prove (4) ; take in () l = 2g, m = 2p 9 with pg-j-i = k -+- and (3 p+ i = A; H- 5, then if E(k) > and JB(«r -+-&)>(>, r=l, 2, 3,...,ii, and amp -<7r, j?>g-hl B(p t ) > 0,.< =, 2, 3,..., q +, p,,..., ^ ^ ^ ^ ^ * ^ xl{ ; < )dk -l # ^^.(g P

13 ) INTEGRAIS INVOLVING PKODUCTS OF ÜT-FUNCTIONS, ECC. 545 X (6), as (4). (4) is proved. The proof of (5) can be deduced from (i) 5. Particular cases : In (4), take p = q z= 0 ; then from the formula... (6), if -(7), which is in WATSON' S book [3] p. 435 formula (5).

14 546 F. M. RAGAB Also, (5) gives when p =z q = 0, amp z < u, Rjk) > «, In (4), write x for # where a; is real and positive, take p = 3 ith a 3 = k -H ; then if Ri*! +- AÏ) > 0, i?(a 2 -H &) > 0, g oo ƒ X*"^( ai, a, ; ; X) oi^5 j ; (a, -H k), l (a, + fc), g (a, +- fc -+- ), 0 n(«,-h &H- ), fc + ; In (5), write x for 0 where # is real and positive, take p = will oc 3 = k -+- ; then if B( ai -+- k) >, I.R(a fc) >, i^(fc) > g g(a x -f- fe- ), ^(a s + fc - ), J (a, -H fc), ), fc; -j,... (20). In (9), write (2X) for and take a =g-f-*i, a. i =:- - n; then from (7) il B(fc)>, jb^fc -H i ± M) > 0,

15 INTEGKALS INVOLVING PRODUCTS OF?-FUNCTIONS, ECC. 547 x r»iâ(i- Hn + k )' (i. + n + fc )> g( ~ n + k )' 2 ^ ^^ / \ / \ \^ I \ à I In (20), write 2X for X and take oc = - H~ «,, oc 2 ^n ^ n; then from (7) ; if J2 ffc-4- g ±»J >, iï(fc) > ^ fc -i- n j rf*; «gj T(fc)xi fc ~i In (9), take «= 0 k'-t-m', a. t = ç> k' m'; then from (9)

16 548 F. M. BAGAB 2fc'-i-2m'-i-2& 2k' 2m'+ 2k 3 2k'-+- 2m' -h 2k 0 s ( ' 4 ' 4 X jn ). 3 2k' 2m' H- 2k X 2, * + i; **r (H-fe' + w' + ^r^-fc'-w' + fc In (20), take ^ = «k' H- m', 4 4 'nl/(?)]^[n^)j' (23)> a 8 = ^ fc' m'; then if (9) gives X 0 2k H- 2m' 2k' 2& 2m' 2&' 2A; H- 2m' 2k', X, k; - In (4), take p = ^rith ai = a, g 0; then if z = ce, where 05 is real and positive, 2(fe)i2( &)> *

17 INTEGRAIS INVOLVING PKODUOTS OF Ê-FUNCTIONS, ECC. 549 In (5), Write x for s Where x is real and positive, take p = with K,=a and# = 0; then if B(k) > g, B(a. k), tatep==o=l; then if p a) > x and a; is real and positive, In (5), takejp = g = l; then if B(k) > ^, i2(p) > 0, iî(fr-t-a) >, iî(3à; -+ a a) > = and as is real and positve - a + -fc,_ a + _ 2 - ft 7rir(7c -4- i)r(g -+- k - i)r(p) i ft_ ± " 4 x«+ fc - ),... (28).

18 550 F. M. KAGAB In (4), take p = 2, p = 0; then if x is real and positive 22(^-4-A:)>0, B(a l H-&)>0, S(&)> o '^i 2 i i. i i ie,: k J d X ^) j,...(29). Similary (5)-witli p = 2, q = 0, gives if as is real and positive, a, -+- k) >, i2(a 2 -+ fc) >, iî(x) > fc l a, k l a, V2 aih ~2 2'2 ai "*"2 ' 2" H 2 2' 2 ) In (4), take p = 2, g =, and get

19 INTEGRALS INVOLVING PRODUCTS OF Ü/-FT7NCTIONS, ECC. 55, where 5(fc) > - is real and positive. In (5), take p~2, q =, and get, - (3), R(p) > 0, B(k -t- a,) > 0, R(k -+- a,) > 0, and x o 2 \k-^l, p / \ - i)r(fc - )( P ) fc_ ,...(32), where JB(X)> =, B(p) > 0, i2(oc l -f- fc) >, JS(a,.-+ fc) > and as is real and positive. BEFEBEKCES [] RAGAB F. M., Proc. Kon. Akad. v. Wet. Amsterdam 57, 44, formula 2, 954. [2] MACROBERT T. M., Functions of a Complex Variable, (4 t?l édition G-lasgow 954). [3] WATSON G. K"., Theory of Bessel Functions, Cambridge 944.

BOLLETTINO UNIONE MATEMATICA ITALIANA

BOLLETTINO UNIONE MATEMATICA ITALIANA S. K. Chatterjea An integral representation for the product of two generalized Bessel polynomials. Bollettino dell Unione Matematica Italiana, Serie 3, Vol. 18 (1963),