However you do it, you get, for any integer k , I C0 (z − z0)k dz = (0 if k 6= −1 i2π if k = −1. The Cauchy Residue Theorem Before we develop integration theory for general functions, we observe the following useful fact. Theorem 2.1. 4. To state the Residue Theorem we rst need to understand isolated singularities of holomorphic functions and quantities called winding numbers. 1. Theorem 2says thatitisnecessary for u(x,y)and v(x,y)toobey the Cauchy–Riemann equations in order for f(x+iy) = u(x+iy)+v(x+iy) to be differentiable. 11.7 The Residue Theorem The Residue Theorem is the premier computational tool for contour integrals. You can compute it using the Cauchy integral theorem, the Cauchy integral formulas, or even (as you did way back in exercise 14.14 on page 14–17) by direct computation after parameterizing C0. Principal part fhas an isolated singular point at z 0, so fhas a Laurent seires f(z) = X1 n=0 a n(z z 0)n+ b 1 (z z 0) + b 2 (z z 0)2 + + b n (z z 0)n + in a punctured disk 0 Resin8 Promo Code 2019,
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