Then for any 0 r can be covered by a finite number of

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Unformatted text preview: 2π. Then z'(t) = ie and so the integral reduces to 2π Û1 Û -it it Ù O dz = Ù e i e dt = 2πi ız ı 0 C Properties of Contour Integrals: Û Û Ù [åf(z) + ∫g(z)] dz = åÙ f(z) dz + Linearity: ı ı C C Û Û Û Ù f(z) dz = Ù f(z) dz + Ù f(z) dz Linearity in C: ı ı ı C#D C D Û ∫Ù g(z) dz (å, ∫ é C ) I ı C 13 Û Û Ù f(z) dz = -Ù f(z) dz ı ı C Creversed Bound for Absolute Value: If |f(z)| ≤ M everywhere on C, then ÔÛ Ô ÔÙ f(z) dzÔ ≤ ML Ôı Ô ÔC Ô where L is the length of C. A simple closed curve is a closed curve with no self-intersections. The domain D is simply connected if every loop can be continuously contracted to a point within D. (Illustrations in class) Theorem 4.2 (Cauchy-Gorsat) If f is any analytic function defined on the simply connected region D and if C is any simple closed contour in D, then Û Ù O f(z) dz = 0 ı C Sketch of Proof:1 We first need a little fact: Fact: Let R be the region interior to a positively oriented simple contour C, together with the points of C i...
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