1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions or li n! ( e )n 1 n-+1!, V'iim ;:; = or as n ---+ oo. This result is known as Stirling' ...

626 Chapter^8 y u -3 -2 -1^0 +1 +2 +3 +4 x (\ -1 -2 -3 (\ -4 Fig. 8.16 vertical asymptotes of the graph of y = I'( x ). By apply ...

· Sequences, Series, and Special Functions 627 The graph of the modular surface of r(z), namely, the graph of v = lr(z)I, is giv ...

628 Chapters or, B(z,( + 1) = ~B(z,() z+.,, if, in addition, z + ( f= 0. (4) For n = 1 we have, using properties 1 and 3, with t ...

Sequences, Series, and Special Functions Hence or Since we obtain or r(x) r(x) (x+e)x <B(x,e)< (e-1)x ex exB(x,e) ex ( x + ...

630 Chapter 8 poles, we may use (8.20-33) to generalize the definition of the B-function to the Cartesian product C x C except a ...

Sequences, Series, and Special Functions 631 This function is a generalization of the usual factorial n! since (1 )n = 1 · 2 · 3 ...

632 Chapter^8 = (a)m+l + ~ (7)(a)m-k+1(b)k ~ (k: 1 )(a)m-k+t(b)k + (b)m+t = (a)m+i + L m ( m k +1) (a)m+t-k(b)k + (b)m+t k=l m ...

Sequences, Series, and Special Functions 633 when Re a, Re b # 0, -1, -2, .... 5. Show that r(z) = I'(z). Also, prove that 7r r( ...

634 Chapters Show that { 0 (^1) xn dx Vi I'((n + 1)/2) lo Vl=X2 = 2 r((n/2) + 1) _ n!! 2 { (n-1)!! '.'.:. if n > 0 is even ...

Sequences, Series, and Special Functions Show that: (a) l(a)nl :::; (lal)n (b) (1 - z)-a = ~ (a)n Zn (lzl < 1) L.. n! n=O 8 ...

636 Chapter^8 In the next theorem we consider some of the most elementary properties of the hypergeometric function. Theorem 8.5 ...

Sequences, Series, and Special Functions = aF( a+ 1, b, c; z) - bF( a, b + 1, c; z) since (a+ n)(a)n = a(a + l)n· 637 Any one of ...

638 Chapters = z ~ (a+ n)(a)n(b + n)(b)n Zn n=O LI n!(c)n = z(O + a)(O + b)w which shows that w = F( a, b, ~; z) satisfies [0(0 ...

Sequences, Series, and Special Functions 639 provided that Rec > Re b > 0. Using (8.22-7) in (8.22-2), we obtain F(a, b, c ...

640 Chapter^8 Because of the symmetry of the hypergeometric function with respect to a and b, we also have F(a,b,c;z)= r(c) 1 1 ...

Sequences, Series, and Special Functions 641 = r( c) r1 th-1(1 - t)c-a-b-1 dt r(b)r(c-b) } 0 = r(c) r(b)r(c - a - b) r(c)r(c - a ...

642 Chapter^8 defining series converges absolutely (by th~ ratio test) for all values of z E C, and so cI>( a, c; z) is an en ...

Sequences, Series, and Special Functions 643 a~ (a+l)n n a ( = - L...J I( l) z =-cl> a+l,c+l;z) c n=O n. c + n c. Hence after ...

644 Chapters where the reversion of the order of summation and integration is justified by the uniform convergence of the series ...