1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

204 CHAPTER 6 • COMPLEX INTEGRATION last expression should equal J: f ( z ( t)) z ' ( t) dt, as defined in Section 6.1. Of cours ...

6.2 • CONTOURS AND CONTOUR INTEGRALS 205 Each integral in the last expression can be done using integration by parts. (There is ...

206 CHAPTER 6 • COMPLEX INTEGRATION Suppose that f (z) = u (z) + iv (z) and that z (t) = x (t) + iy (t) is a parametrization for ...

6.2 • CONTOURS AND CONTOUR INTEGRALS 207 y y (a) The line segment. (b) The portion oflhe parabola. Figure 6.8 T he two contours ...

208 CHAPTER 6 • COMPLEX INTEGRATION y y l+i l+i .+-<1o---+---..._..x I - I (a} The semicirc ular path. (b) The polygonal p ...

6.2 • CONTOURS AND CONTOUR INTEGRALS 209 Using the change of variable t = - r in this last equation and the property that J: f ( ...

210 CHAPTER 6 • COMPLEX INTEGRATION ...

6.2 • CONTOURS ANO CONTOUR INTEGRALS 211 y Figure 6.10 The distances lz -i i and lz +ii for z on C. EXAMPLE 6.11 Use Inequality ...

212 CHAPTER 6 • COMPLEX INTEGRATION (b) C = c, + G2 + Gs, as indicated in Figure 6.12. I • x 3 - 3 -2 -I Figure 6.11 Figure 6.12 ...

6.2 • CONTOURS AND CONTOUR INT EGRALS 213 {b) The contour C that is oriented clockwise, as shown in F igure 6.14(b). y y I x -I ...

214 CHAPTER 6 • COMPLEX lNTECRATION Let z (t) = x (t) + iy (t), for a :$ t :$ b, be a smooth curve. Give a meaning for each of ...

y Interior 6.3 • THE CAUCHY-GOURSAT THEOREM 215 y Exterior -1-----1--4-+.x Figure 6.15 The interior and exterior of simple close ...

216 CHAPTER 6 • COMPLEX INTEGRATION ...

6.3 • THE CAUCHY-GOURSAT THEOREM 217 y a b Figure 6.18 Integration over a standard region, where C = C 1 + C 2 • We are now read ...

218 CHAPTER. 6 • COMPLEX INTEGRATION y d c Figure 6. 19 Integration over a standard region, where C = Ca + C4. We give two proof ...

6.3 • THE CAUCHY- GOURSAT THEOREM 219 y Figure 6.2 0 The t riangular cont ours C and C^1 , C^2 , C^3 , and C^4. ...

220 C HAPTER 6 • COMPLE X INTEGRATION y Figure 6. 21 The contour 0,. that lies in the neighborhood lz -zol < 6. ...

6.3 • THE CAUCHY-GOURSAT THEOREM 221 EXAMPLE 6.12 Recall that expz, cosz, and zn (where n is a positive inte- ger) are all enti ...

222 CHAPTER 6 • COMPLEX INTEGRATION x Figure 6.22 A simple connected domain D containing the simple closed contour C that does n ...

6.3 • THE CAUCHY- GOURSAT THEOREM 223 y (a) The contour Ki and domain Di. (b) The contour K 1 and domain D 2. Figure 6.24 The cu ...