1550251515-Classical_Complex_Analysis__Gonzalez_

Integration 485 Show that the substitution z = ei^9 , 0 :::::; () :::::; 211", transforms the real integral into the complex i ...

486 Chapter 7 14. Let C be a simple closed contour and suppose that f is analytic on C U Ext C, and that limz-+oo f ( z) = L. Pr ...

Integration 487 y* Fig. 7.26 where 1 is a rectifiable arc, not necessarily closed or simple, z E C - 1* and f is continuous alon ...

488 Chapter^7 = J J(() d( + J hf(() d( (7.20-3) ((-z)^2 ((-z-h)((-z)^2 'Y 'Y Since f is continuous on 'Y, there exists a constan ...

Integration 489 [! J(()d( J f(()d( l 'Y ((-z-h)n((-z) - 'Y ((-z)n+l [! J(()d( J f(()d( l + 'Y (( - z - h)n-1(( - z)2 - 'Y (( ...

490 Chapter 7 Proof We have F(n)( z ) = n. I J (( f(() - z)n+l d( 'Y for z E Ro and n = 1, 2,.... This formula applies also to F ...

Integration 491 f(n)(z) = ~ J J(() d( 27ri ((-z)n+l a+ (7.21-2) These formulas give a representation in terms of integrals of th ...

492 Chapter^7 within C. Also, 7r lies inside C. Hence, by using (7.21-2) with n = 3, we get J t~n: :~ = 2;i !"' ( 7f) = ~i ( - c ...

Integration 493 4. Evaluate fc zP(l - z)q dz, where C: z = reit, 0 :::; t :::; 271", 1· > 1, and p, q are integers. Consider ...

494 The Laguerre polynomial Ln(z) is given by Ln(z) = ez Dn(zne-z) Apply (7.21-2) to show that I 1 (n -((-z) Ln(z) = 2n.. (( e ...

Integration 495 small enough), and similarly, F+(z 0 ) denotes the limit of F(z) as z -t z 0 from the right along a nontangentia ...

496 Chapter 7 Proof By (7.21-2) we have j<n)(zo)= ~ J f(()d( 21l"i ((-zo)n+l c+ This formula holds also for n = 0 with the no ...

Integration 497 Note As we have seen in Section 5.14, in the special case of a polynomial of degree n ;::: 1 the inequality (7.2 ...

498 Chapter^7 for lzl > R. On the compact set lzl SR the real continuous function lg(z)I is bounded: i.e., there is f{ > 0 ...

Integration 499 0 A Fig. 7.27 If we now let f(a) = b, then f will be defined and analytic in all of A. Definition 7.12 A point s ...

500 Chapter^7 1. f is defined for ( on a contour C and for z in an open set A: i.e., f is defined on C x A. 2. lf(z,()I :5 M (a ...

Integration 501 Now choose h such that ihl < 1/ 2 r. Then iu -zl =rand iu -z - hi>^1 / 2 r. From (7.27-5) we have F(z+h) = ...

502 Chapter 7 the real part of the function on the total boundary of the region. In this section these results are established f ...

Integration R^2 - r^2 {2" d'lj; = 271" lo J(() !( - z!^2 since ( z (( -zz R^2 - r^2 ( - z + ( -z = !( - zl^2 = !( - z!2 Next, ad ...

504 Chapter^7 (( - z)(( - z) =I( - zl^2 Corollary 7.18 As a by-product of the foregoing proof we have formu- las (7.28-8) and (7 ...