1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

224 CHAPTER 6 • COMPLEX INTEGRATION t Corollary 6.1 Let zo denote a fixed complex value. If C is a simple closed contour with po ...

6.3 • THE CAUCHY-00URSAT THEOREM 225 Figure 6.23. Then the contour C = C 2 - C 1 is a parametrization of the boundary of the reg ...

226 CHAPTER 6 • COMPLEX INTEGRATION The points z = ±i./2 lie interior to ct (0), so Corollary 6.1 implies that --= =21Tt. 1 dz. ...

6.3 • THE CAUCHY- GOURSAT THEOREM 227 y y (a) The figure eight contour C. (b) The contours C 1 and Cz. Figure 6.28 The contour C ...

228 CHAPTER 6 • COMPLEX INTECRATION Show that fc z-^1 dz = 27ri, where C is the square with vertices 1 ± i and -1 ± i and havin ...

6 .4 • THE FUNDAMENTAL THEOREMS OF INTEGRATION 229 Evaluate fct(o) lz l^2 expz dz. 12. Suppose t hat f (z) = u (r , 9) +iv (r, ...

230 CHAPTER 6 • COMPLEX INTEGRATION Figure 6.30 The contours C1 and C2 joining zo to z. ...

6.4 • THE FUNDAMENTAL THEOREMS OF INTEGRATION 231 y Figure 6.31 The contours r i and r 2 and the line segment r. R em ark 6.2 It ...

232 CHAPTER, 6 • COMPLEX INTEGRATION Theorem 6.9 gives an important method for evaluating definite integrals when the integrand ...

6.4 • THE FUNDAMENTAL THEOREMS OF INTEGRATION 233 y = x (a) The path C joining z 1 and Z2· = (b) The path that is a portion of t ...

234 CHAPTER 6 • COMPLEX INTEGRATION f c cos z dz, where C is the line segment from - i to 1 + i. fc expz dz, where C is the lin ...

6. 5 • INTEGRAL REPRESENTATIONS FOR ANALYTIC FUNCTIONS 235 6.5 INTEGRAL REPRESENTATIONS FOR ANALYTIC FUNCTIONS We now present so ...

236 CHAPTER 6 • COMPLEX INTEGRATION Figure 6.33 The contours G and Co in the proof of Cauchy's integral formula. • EXAMPLE 6. 21 ...

6.5 • INTEGRAL REPRESENTATIONS FOR ANALYTIC FUNCTIONS 237 f (z) = ex~~;z) and use Theorem 6.10 to conclude that r exp(i?rz)dz 1 ...

238 CHAPTER 6 • COMPLEX INTEGRATION EXAMPLE 6.24 Let z 0 denote a fixed complex value. Show that if C is a simple closed positi ...

6.5 • INTEGRAL REPRESENTATIONS FOR ANALYTIC FUNCTIONS 239 We now state two important corollaries of Theorem 6.12. t Corollary 6 ...

240 CHAPTER 6 • COMPLEX INTEGRATION 7. Find fct(o) z -^3 sinh (z^2 ) dz. Find fcz-^2 sinz dz along the following contours: (a) ...

6.6 • THE THEOREMS OF MORERA AND L!OUVILLE, AND EXTENSIONS 241 Use Cauchy's integral formula to show that 1 r (e- 1rd~ P.,. (z) ...

242 CHAPTER 6 • COMPLEX INTEGRATION with center zo, then we can show that the value f (zo) is the integral average of the values ...

6.6 • THE THEOREMS OF MORERA AND L IOUVILLE, AND EXTENSIONS 243 Figure 6. 34 The "chain of disks" Do, D1, ... , D,. that cover C ...