By A. M. Mathai, R. K. Saxena
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Additional info for Generalized Hypergeometric Functions With Applications In Statistics And Physical Sciences
Example text
Ii) v _> 2 and arg z __> (v - I)~ . (iii) v > 2 * and srg z < -(v - I)~ . , in the following cases: The results of the theorem are not applicable when q ~ p+2 with v and -(v*-l)~ ~ arg z < (v*-l)~. 4. 12) the following interesting particular cases of the theorem are obtained. 2). 1). 1). 4). 11) w i t h q = p+2. 3) w i t h q = p+2. 6. i < m < p+l. 17). A l s o (v * 3 "~ 1 + 2k- ~ ) ~ _< srg z < (v ~ + 2~ + ~)~. T h e n the following are the a s y m p t o t i c expansions of the f u n c t i o n m~n Gp,p+l(Z) for large values of P "~ Z exp (-in(2k + l)at} t=l m~n Gp,p+l(Z) (i) z x Tm , n (t "" .
Ap-l'ap-r Gm'n( ! P ] P q x P,q xl b )~ = x q b 1 .... ,b q and r is any positive integer. 9 Show that P It r(b. ) a j=l p+l p+l z) p+iFp ( b ; P ] 11 r(aj) j=l p+l p+l x Z k=l P(ak ) P(aj-ak) (ei~/z) ak j=l P r(bj-a k) j:l -40x l+ak-bl .... 'l+ak-bp p+iFp(l+ak_al ' where 0 < arg z < 2~. ,l+ak-ap+ I i) ; z Also prove that this formula provides the analytic con- tinuation of the function p+iFp on the left from inside the unit disc to the outside of this disc. i0 Establish the following expansions for the G-function.
___~3 G+v+3 , z__ ). 6. -~+v+l ~, G~v+l r (--i---)r < ) ~I Jv (z) - cos( ~ )~I Yv (z) ] " Elliptic Functions and Integrals Complete Elliptic Integrals. 7 = I f (l_k2sin20)2 of ! 2FI(" i !. l;k 2) , 7'2' =2 Theta Functions. 1 7 O0(Vl'r) = (-iz)- OI(VIT) 1 (v- ~ +r) 2 ~ exp [-i~ co r~ 1 o~ = (-iz) -2 Z r ~ z ] " 1 2 [ - i ~ ( v + r - 3) (-I) r exp co 1 • T oo 02(Vl~) (-iT) -I/2 Z (-l) r exp[ -i/2 -i~(v+r)2z oo _i~
