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Prove this Theorem 1.5. (Cauchy-Goursat Theorem for Multiply-Connected Domain) Let C be a simply closed contour and let C1, C2,... , Ck be simple closed contours in the interior of C such that 1. C;N C; = ¢,i + j; 2. Int(C:) n Int(C;,) = 4, i ± j. Let R be the region interior to C but exterior to each C,, j = 1,2, ... , k. Then, R is a multiply connected domain. Let B be the boundary of R oriented positively, then | f(2)dz = [ 5(2)dz +£ /, s(-)dz = 0. %3D i=1 Note that f + f +

Prove this Theorem 1.5. (Cauchy-Goursat Theorem for Multiply-Connected Domain) Let C be a simply closed contour and let C1, C2,... , Ck be simple closed contours in the interior of C such that 1. C;N C; = ¢,i + j; 2. Int(C:) n Int(C;,) = 4, i ± j. Let R be the region interior to C but exterior to each C,, j = 1,2, ... , k. Then, R is a multiply connected domain. Let B be the boundary of R oriented positively, then | f(2)dz = [ 5(2)dz +£ /, s(-)dz = 0. %3D i=1 Note that f + f +

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section: Chapter Questions
Problem 12T
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Prove this Theorem 1.5. (Cauchy-Goursat Theorem for Multiply-Connected Domain) Let C be a simply closed contour and let C1, C2, ... , Ck be simple closed contours in the interior of C such that 1. C¡NC; = ¢, i # j; 2. Int(C:) N Int(C;) = 0, i + j. Let R be the region interior to C but exterior to each C;, j = 1,2, .. , k. Then, R is a multiply connected domain. Let B be the boundary of R oriented positively, then k f(-)dz = | f(2)dz +E f(2)dz = 0. i=1 Note that + f.
Transcribed Image Text:Prove this Theorem 1.5. (Cauchy-Goursat Theorem for Multiply-Connected Domain) Let C be a simply closed contour and let C1, C2, ... , Ck be simple closed contours in the interior of C such that 1. C¡NC; = ¢, i # j; 2. Int(C:) N Int(C;) = 0, i + j. Let R be the region interior to C but exterior to each C;, j = 1,2, .. , k. Then, R is a multiply connected domain. Let B be the boundary of R oriented positively, then k f(-)dz = | f(2)dz +E f(2)dz = 0. i=1 Note that + f.
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