15 4 if f is analytic throughout a simply connected

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Unformatted text preview: œ, and |z-zi| ≤ diamSi. This gives Û Û Ù Ù |O f(z) dz| = |O ©(z)(z-zi) dz| ı ı Si Si ≤ œ ¥ diam Si ¥ length Si ≤ œ 2 si ¥ 4si in the case of squares totally inside R = 4 2 œ¥Area of Si or ≤ œ ¥ 2 si ¥ [si + length (Ci)] = 2 œ(Area of Si + siLength(Ci)] where si = length of an edge in Si and Ci is the portion of C inside Si. Adding these up gives a total not exceeding 4 2 œ¥Total area of R + 2 œ ¥ Total area of R + 2 œ(S¥Length(C)] where S is the length of some square that totally encloses R . Now, since is arbitrarily small, we are done. Consequences: 1. If f is analytic throughout a simply connected region R containing two nonintersecting contours C and D with the same endpoints, then Û Û Ù f(z) dz = Ù f(z) dz ı ı C D 2. If R is any old region (not necessarily simply connected) and C and D are closed simple contours with C enclosing D, such that the region in between C and D is simply connected, then Û Û Ù Ù O f(z) dz = O f(z) dz ı ı C D 3. If...
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This document was uploaded on 03/20/2014 for the course MATH 144 at Hofstra University.

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