1550251515-Classical_Complex_Analysis__Gonzalez_

Integration 445 Fig. 7.8 If z + h is another point within the same disk, we have 1 z+h F(z+h)=<n zo J(()d( where r consists o ...

446 Chapter^7 which shows that Iimh-+o[F(z + h) - F(z)]/h = J(z). Remark Obviously, in Lemma 7.2 the disk lz -z 01 < r can be ...

Integration 447 Fig. 7.9 n-1 j f(z)dz = L j f(z)dz '"'I k=O'"'lk and if Fk is a primitive off on the disk Dk, we obtain n-1 j f( ...

448 Chapter 7 Now any two admissible sequences s 1 and s 2 have a common refinement sa = s1 U s 2. Since the result agrees for s ...

Integration 449 = J P 6..x + Q 6..y = 0 8Lmn Corollary 7.11 If 6..xQ - 6..yP = 0 on Lm-l,n-1, then f P 6..x + Q 6..y = 0 JaLmn · ...

450 Chapter? If we put j k Zjk = h( N ' N ), j, k = 0, 1, ... 'N j j+l k k+l Sjk = [ N , JV] x [ N , 1\T ], j, k = 0, ... , N - ...

Integration 451 = J f(z)dz (7.10-12) h(8S) Hence, by Corollary 7.11, we get, in view of (7.10-11) and (7.10-12), J f(z) dz= 0 h( ...

452 Chapter^7 In what follows we state as corollaries some particular versions of the Cauchy-Goursat theorem that are often foun ...

Integration 453 instance, as seen in the example following Corollary 7.4, we have f(z~za)n=O· c for n = 2, 3, ... , C: z-a = rei ...

454 Chapter^7 approaches fc f(z) dz as the norm of P tends to zero. Because C and Cr are homotetic curves, the sum n L J[b + rp( ...

Integration 455 Remark If the region R does not contain a point b such that every ray with origin b meets C at just one point, i ...

456 Chapter^7 c Fig. 7.1:~ The points A' and H divide the contour C into two arcs C' and C", both with the. same orientation as ...

Integration 457 Hence (j + j) f(z)dz ~ (i +£ + 1+[,+1 +_[, f + j, + f + j, + f + j) f( z) dz ~ 0 which gives J f(z)dz+ J f(z)d ...

458 Chapter^7 R c Fig. 7.13 a denotes some point inside Ci, we can write (7.12-2) in the form J f(z)dz = no1(a) J f(z)dz = nc(a) ...

Integration 459 arbitrarily closely by a piecewise regular contour, namely, an inscribed polygon. To justify the definitions abo ...

460 Chapter^7 Proof Choose a 0 r/. C* and € > 0. We must show that there exists 8 > 0 such that la -ao I < 8 implies th ...

Integration 461 following manner: If for any point a E Int C* we have ilo( a) = + 1 the contour C is described once in the posit ...

462 Chapter 7 -1or0, so that C 1 ,...., C 2 in R but C 1 f C 3 in R. If f is analytic in R, then J f(z) dz= j J(z) dz 01 02 At t ...

Integration 463 Consider the closed contour C 1 = ADBCDACBA. Since in every inte- gral along C 1 the integral over AD cancels wi ...

464 Chapter^7 Proof We have J f(z)dz = t J f(z)dz G k=l'Yk where each /k is described according to the direction and number of t ...