Prove thisTheorem 1.5. (Cauchy-Goursat Theorem for Multiply-ConnectedDomain) Let C be a simply closed contour and let C1, C2, ... , Ck be simpleclosed contours in the interior of C such that1. 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 Roriented positively, thenkf(-)dz = | f(2)dz +E f(2)dz = 0.i=1Note that+f.

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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 +

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.

Question

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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