1550251515-Classical_Complex_Analysis__Gonzalez_

Integration 465 where '"'ft is a circle with center ak and sufficiently small radius rk described once in the positive direction ...

466 Chapter^7 y -u + ia ia z = x + ia u + ia z = -u + iy z = u + iy z = x -u 0 u x Fi.g. 7.l'T Decomposing ('/.15-1) into four i ...

Integration 467 converges, and we get or (7.15-4) The same result is obtained if a is supposed to be negative, since cos 2ax is ...

468 Chapter^7 Fig. 7.18 where BM P and P NB are the two arcs into which the contour el (suppos- edly described once in the posit ...

Integration 469 If the connectivity of G is p > 2 and the contours C1, C 2 , ••• , Cp-l are not null homotopic in G, then if ...

470 Chapter^7 the value of the integral along a closed contour C 1 around the origin, de- scribed once in the positive direction ...

· Integration 471 so that F 1 (z) = Logz + K for z E G - L 0 , K being a constant. But F1 (1) = 0 and Log 1 = 0, hence J{ = 0, a ...

472 Chapter^7 Let ][' = l:;=l ArCr. where the cycles Cr are closed contours. Prove that n ilr(a) = LArilcr(a) r=l and show tha ...

Integration 473 Study the multiple-valued function defined by F( ) = ("Y) 1z _!:L z 0 1 + (2 in C - {i, -i}, and show that F(z ...

474 Chapter^7 have, by (7.12-2), _1 J J(()d( = _1 J J(()d( 27l"i ( - z 27ri ( - z C C1 = _1 j f(z)d( + _1 j f(() - f(z) d( 27l"i ...

Integration 475 case in which D is starlike, we may proceed as in the proof of Theorem 7.14 to obtain this result. Cauchy's form ...

476 Chapter^7 (; 2 3 x c Fig. 7.21 =^2 7ri 2i(i i_ 3) +^2 7ri -2i( =;-3) = - ~ 7ri The following are generalizations of Theorem ...

Integration Fig. 7.22 But c J f(()d( = 0 (-z 'Yl (r = l, ... ,p) by Theorem 7.23, and ~ J f(()d( = f(z) 27ri ( - z -y+ 477 by fo ...

478 Chapter^7 C +(-Ci)+···+ (-Cn)· If f is of class C^1 (R) and z ER, then Or(z)f(z) = ~ J f(()d( - Or(z) jrf J, ded71 (7.17-6) ...

Integration 479 Thus if we let p ---+ 0 in (7.17-7), we get 2i ff ( ~ z de dry= f {~~ d( - 27rif(z) ft r+ or f(z)= ~ff(() d(-~ f ...

480 Chapter^7 the local interior angle at z 0 , denoted >.(z 0 ), is defined to be 7r (at such a point Chas a unique tangent) ...

Integration By Corollary 7.13 we have or f(() d( = 0 (-zo f(()d( - ~ J f(()d( ( -zo - 27ri ( - zo · r(r) = f(zo) J 27ri --+-d(^1 ...

482 Chapter 7 for JzJ large enough, where A > 0 and m > 0 are real constants. Then if b > a and Re z > b, we have .f ...

Integration 483 For ( E C 1 and .>. sufficiently large, we have A 2A = IClm+lll - z/(I < .>.m+i since ICI ;::: .>.fo ...

484 Chapter^7 3. Suppose that f is analytic on and within the simple closed contour C, and .z f/. C*. Evaluate the following. (a ...