INTRO COMPLX VARIAB
INTRO COMPLX VARIAB MATH 122A
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This 1 page Class Notes was uploaded by Estevan Champlin Sr. on Thursday October 22, 2015. The Class Notes belongs to MATH 122A at University of California Santa Barbara taught by Staff in Fall. Since its upload, it has received 27 views. For similar materials see /class/226884/math-122a-university-of-california-santa-barbara in Mathematics (M) at University of California Santa Barbara.
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Date Created: 10/22/15
Math 122B Complex Variables The CauchyGoursat Theorem CauchyGoursat Theorem If a function f is analytic at all points interior to and on a simple closed contour 0 ie f is analytic on some simply connected domain D containing 0 then f2 d2 0 0 Note If we assume that f is continuous and therefore the partial derivatives of u and v are continuous where f2 u7y iv7y7 this result follows immediately from Green7s theorem Letting R be the region enclosed by the curve 07 Cf2d2Auayivz7ydaidyAudxivdyi vdxudy AivzfaydAiAuzivydz40 since f is analytic use the Cauchy Riemann equations However7 the Cauchy Goursat theorem says we don7t need to assume that f is continuous only that it exists Theorem An extension of Cauchy Goursat ff is analytic in a simply connected domain D then Cf2d2 0 for every closed contour C lying in D Notes 0 Combining this theorem with Theorem 427 every function f that is analytic on a simply connected domain D must have an antiderivative on the domain D 0 Given two simple closed contours such that one can be continuous deformed into the other through a region where f is analytic7 the contour integrals of f over these two contours have the same value In other words7 f might not be analytic in some region R7 but if it is analytic outside of R7 then the value of the contour integrals off must be the same for all closed contours that enclose R 7 of course7 this value doesn7t have to be 0 since f is not analytic everywhere See the corollary below
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