f(x)=f x/:x~,j~(xy)f(y) dy, R(v) > (1-1) O

Size: px

Start display at page:

Download "f(x)=f x/:x~,j~(xy)f(y) dy, R(v) > (1-1) O"

Darlene Oliver
5 years ago
Views:

1 SME NEW KERNELS FR THE DERIVATIN F SELF-RECIPRCAL FUNCTINS BY H. C. GUPTA, PH.D. (Christ Church College, Cawnpore) Received March 14, 1945 (Communicatcd by Prof. K. C. Pandya, F.A.SC.) w 1. INTRDUCTIN FLLWING Hardy and Titchmarsh we shall calla function R~ if it is self-reciprocal in the Hankel transform of order v, that is, if it satisfies the integral equation f(x)=f x/:x~,j~(xy)f(y) dy, R(v) > (1-1) A number of rules for the derivation, from a known R~, function, of another self-reciprocal function of a different order, v, say, has been gÿ from time to time by several authors.* ne such rule is that iff(x) is R~,, then the function is R, provided that c,~co g (x) = f P (xy)f(y) dy (1-2) P(x) = ~-~ 1 f 2 s 1" ( " ( F (s) x -~ ds, (1.3),J r --to~ where F (s) -- F (1 -- s). The symmetry in q and v of the integrand of (1.3) shows that if f(x) is R~ then g (x) is R~. The function P (x) is ealled a kernel for transforming an R~, into ah R~ and rice versa. We shall indieate this by saying tlaat the kernet is C (/z, v). In the present papera few kernels have been investigated by this rule and employed to derive some new R~ ftmctions. * E. C. Titchmarsh, Theory of Fourier Integrals (xford, 1937), ~ Besides the references given there on p. 267, see W. N. Bailey, Jour. London Afath. Soc., 1931, 6, a.ad B. Mohan, Proc. Physico-lWath. Soc. dapan, 1936, 18 (3), ; Quart. Jour. of Ma;h., 1939, 10, (40), , lndian Jour. of Physics, 1941,

2 Some New Kernels fiar Derivation l c Self-Reciprocal Funclions THREE KERNELS F THE TYPE X a In (89 K,,, (89 The limiting case of MacRobert's integral~ may be put in the form f _ x~-lx x I,, (89 K., (89 dx = F {89(1 -- A-- s)} /' ( r'z {89(n A 4- m s)} 3/rr 2 =-x~/'~ {89(n -- )t-- s 4- m) 1} where R (-- n 4- m) < R (;~ s) < 1. Since the integral is absolutely convergent, it follows by Mellin's Inversion Theoremw that e t ~ x x I,~ (89 Kn, (89 = 2-~. (89 F(89 v F( F(s) ds, (2.1) where c -- i _r' {89(1 -- a-- s)} _r' { (89(a n 4- m s)} 2 x-2 F (s) - r~ {89(n - a m - a) 1 ) V {89(,,, 89 V {89(~, 89 ~/,~ n comparing with (1.3) we find that the function on the left of (2.1) is a kernel if F (s) - F (1- s), which requires that the parameters --89 )~, 89 ~, v, l~=n 4- m 4-(89 ) 0 3n n-- 89 severally but in any order, This admits of three valid solutions : (i) m = n = - ;~; /, = 89 3n, v n-- 89 yielding a kemel of the class Cts. ~ ~ namely, x-" In (89 K,, (89 (ii)), = 0, m = n, tz = 2n 89 v = (iii) m = 89 namely, I,, (89 K~ (89 89 > R (n) > ; 2n yielding a C (2n 4-89 namely, R (n) > n=~~v,?,=~-~v, ~=vl, yielding a C(v, vl), -~ 0 ~) --89 x e I~~, (89 R (i,) > -- lo T. M. MacRobert, Quart. Jour. of NIath., 1940, 11, 98. For brevity/"~ (,x a- v) is written for ff (,X v) F' (a -- v). G. H. Hardy, 5r of M'ath., 1918, 47, t The conditions necessary for the validity of the kernels ate obtained from the considera. tion that R~ functions are defined cnly for R (y) > --1.

3 230 H. C. Gupta 3. APPLICATIN F TttESE KERNBLS T THE R~ FUNCTIN* x't e-t x' T, 2t~ (x2),p= o R A VE INTEGER For this purpose we first evaluate the general integral andt I= f I. (89 Kra (89 (xy) x y~'89 e- ~'2 T~, 2p (y2) dy. )~(-x)~ Since T,," (x) = ~=o~ (n9 - r l-~r. I. F (v 1 r) ' K, (z)= 89 F (1,) F(1 -- 1,) {I_. (z)- I~ (z)} Ix (2z) I~, (2z) = F (~ 1) zx~' F (tz therefore 1)2F3 (1~ (1 ~, 1 )~/~), ~, 1 1 ~ 89 (~ tz; tz); 4z 2) xx ~" ( --)" I-- 2-F(n 1),=o r!(2p--r)! P(I f vr) yx, 8, ~ e" 89 X 0 a similar function I { F(m) ( (88 2F3 k 1 n, 1 - m, 1 n - m ; with dy. - m written for m (3.1) Now by using the equivalent infinite series for the hypergeometric function 2Fs and integrating term-by-term by means of the formula f y2,,,1 e--~y ~ dy = 2 m F (1 m), R (m) > - 1, o we have i= ~ [(-)".~X/r[ (2p- r)! F(I v r) l-(1 n).2 ~n~'89 " t~---0 F(k r)f(m) (89 -m 1), 1 89 k r ;~ a similar term with] (~~2-~-2)-~- zf3 kl n, 1 - m, 1 n - m; 89 2 ] - m written for mi = (x), say, where 2k = )~ v n - m ~,3 R (k) > 0, and R (m k) > 0. The term-by-term integration effected in (3.1) is justi since the 2F3 is an integral frunction of y and its equivalent inflnite series is therefore uniformly convergent in any arbitrary interval (0, ct) and the remaining part of the integrand is positive, bounded and integrable in (0, a) and the complete integral I converges under the conditions stated. * B. M. Wilson, llr of Math., , 53, t (3. N. Watson Theory ofb~ssel Functions (Camb., 1922), pp. 78, 147. henceforth be referred to as B.F. This treatise will

4 Some New Ker~~els fox" Do'ivation o1" SellZRedpyoca! Functions 231 We may now carry out the application of the various kernels by mere substitution in the final vatue r (x) of I; but before we do so we have to ascertain the asymptotic behaviour of I in order to test the validity of the R, functions f(x) by the convergence of the integral (1.1). For this we have recourse to the Mellin-Barnes type of contour integral for r (x). It is ~" &r xnx-,~ ; I'(n-- m 14-2s) F(r k s) -r'(-- s) I'(m-- s) (~ 2)" da",=o /'(n4-1-t-s) _r'(n--m ls) r where A r is a constant independent of x~ We next alter the contour of integration so as to gÿ us the asymptotic behaviour of ~ (x). Carrying out the usual analysis we find that the leading terms in the asymptotic expansion are ax 4- bx, a, b being numerical constants. Evidently there ate two ways of applying each kernel. Thus in the case of the first kernel we may put in ~91 (x), m--n=- ;~ and either v=3n4-89 or v=-n--=89 obtaining an R_,,_ t and an Ran89 function respectively. For instance, the former is z~' {( -- 2)'/r! (2p -- r)! _r' (r.~ n ~)} [P (n) /" (n r 1) (2v'2/x) '~ r= ::=F= (89 nr l; 1 n; 89 I'(14-2nr) (x/2~/2) '~ x2f2(n89 ln,l2n; (3-2) The R, functions investigated in this way are all in the form of finite series of hypergeometric functions of the type ~F~ (892) obtained by the first two kernels and of the type Fa obtained by the third. These reduce to a single pair of hypergeometric functions when p= 0 and these pairs in some cases are expressible in terms of the more common functions. Thus for instance (3.2) reduces when p = 0 to x "-1 W_~,~, -in (89 x2) ei":", 89 > R (n) > -- 1, which is a special case of an R, function given by Bailey.* 4. Two I~RNV.LS 1~ THE T'91 x x J, (x/x/2) K~ (x/~/2) We shall first derive by the aid of Mellin's inversion theorem from the rule of w 1 a functional equation satisfied by infinite integrals of the kemels multiplied by x ~-1. Since P(~ i~t)--e-~,~,~~ t t"-~ as t ~,,~ and s= c it * W. N. Bailey, Jour. London Math. Soc., 1930, 5, A3

5 232 H.C. Gupta tt follows that as I ti -->, (s)= r( r ( ~ ~-,', t~c~,,-l>~. Consequently supposing F (s) to be analytic and o {e(~"~)q ' :, 0}, c to be real, and > 0 we see that ~ (s) F (s) --0 uniformly as I t [ ~ co and the integral f ~ c o I~(cit) F(cit)ldt eonverges. Hence by Mellin's inversion theorem we have from (1.3) co f(s) -- f x r-~ P (x) dx= 2" P ( P ( F (s) ds and the relation F (s)---f (l--s) assumes the forro of the functional equation 2t~F( f (s) - 2s/'( f (1 -- s) (4) showing that either side isan even function of s--89 The functional equation (4) is at times more convenient and serviceable to use for the investigation of kernets os illustrated below. The special case /~= v and a= b of the formula (1), B.F., p. 410, can be put aftera slight change of the variable in the form o /" (lp-- L A- 88 1) (2 V'2) ~a-'' R (s ~ p) > I R (,o) I. In order that this might give us a kemel, the functional equation to be satisfied is 4 (s) $ (1 -- s) = $ (s) $ (1 -- s) where ~(s) ~ v( V(89 89 v( ~) and (s) ~ v( v( The foilowing solutions ara possible :-- (i) ~= 89, p = 89 yielding a C (0, 2v), namely, X (ii))~--- {, ~ = 2, P = 89 yielding a C (2, 20, namely, ~. (~) ~,..~.(~~, ~.~ > -~.

6 Some New Kernels for Deriva~ion of Sel/:Reciproca91 FuncEons APPLICATIN F THE KERNEL$ F w 4 T THE R~ FUNCTIN x~ J~ (89 2) Taking the corresponding R function for the apptication of the first kernel and making use of the integral* r f 2y Jr (yx/2) K~ (yv/2) Jo (Y~[ 2~) dy = ~ I89 (~) K.~~ (~) (5,1) we are led to the R r function a result due to Erdelyi. ~/x I~r (88 x2) K88 (88 R (v) > - 1, Applying the seeond kernel to the corresponding R~ function we obtain the R~ r funetion g (x)= f (xy) ~ Jr (~22) Kr (~2) y~ Jx,89 41 To evaluate this integral we differentiateÿ both the sides of (5.1) with respeet to ~ and use the recurrence formul~e (B.F., w 3-71) and the formula and obtain finally the R r function Ir (z) Krx (z) Irt (z) K~ (z) = l lz g (x)= (1-89 x ~ I~r (88 K~r (88 x 89 ~ I,r-1 (88 K,r (88 *) If we use the asymptotic expansions (B.F., w 7-23) and 4~x [ 4v~'--l (4v~--- 1) (4v~--- 9) ] Kr (x) ~ e ~ 1 --8~---- 2! (8x)2 "" ex [ 4v~--1 (4v~---1)(4v2--9)_ ] Ir(x)-~ 1-- 8x 2!(8x) 2 "'" we find after some ealeulation that the leading term in the asymptotie expansion of g (x) is x "~sz. Consequently g (x) is R r if R (v) > We eonelude by giving two other kemels investigated by this method. They are of the elass C (v, u 1) valid for R (v) > - 1. (i) x '~ et* Wa,,,-,~4.,~ (x), (ii) x~89 e-* T~~, (2x). Two special eases of (i), viz., for m = are x "i e ~x D_~_ra (a/2-x) and x ~ ev D-4r-2 (a/ux). * S. C. Mitra, Bulletin Catcutta Math. Soc., , 25, 89. ~: A. Erdelyi, Jour Lona. Math. Soc., 1938, 13, 153. "~ A pror easily justifir by the uniform and absolute convergence of the integrals involved.

QUADRUPLE INTEGRAL EQUATIONS INVOLVING FOX S H-FUNCTIONS. 1.Dept. of Mathematics, Saifia Science College, Bhopal, (M.P.), INDIA

QUADRUPLE INTEGRAL EQUATIONS INVOLVING FOX S H-FUNCTIONS Mathur 1, P.K & Singh 2, Anjana 1.Dept. of Mathematics, Saifia Science College, Bhopal, (M.P.), INDIA 2.Dept. of Mathematics,Rajeev Gandhi Engineering