Theorem 5.9 Cauchy's Integral Formula Suppose that f is analytic in a simply connected domain D and C is any simple closed contour lying entirely within D. Then for any point zo within C, 1 f(z) f(zo) = i fo dz. (1) 2πi 2-20 In Problems 1-22, use Theorems 5.9 and 5.10, when appropriate, to evaluate the given integral along the indicated closed contour(s). 1 21. fc 2³ (z − 1)² dz; |z2| = 5 Theorem 5.10 Cauchy's Integral Formula for Derivatives Suppose that f is analytic in a simply connected domain D and Ci any simple closed contour lying entirely within D. Then for any point z within C, n! f(") (20) = 2 m i f (z −²20)²+1' f(z) dz. (6 2πi

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