9 cauchy integral formula

# 9 cauchy integral formula - ANTIDERIVATIVES Let f(z) be...

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ANTIDERIVATIVES Let f(z) be continuous function in a domain D. If there exists a function F(z) such that

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then F(z) is called an antiderivative of f(z) in D. D, in z all for ) ( ) ( z f z F =
Remark1: An antiderivative of a given function f is an analytic function. Remark 2: An antiderivative of a given function f is unique except for an additive complex constant.

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Theorem: Suppose that a function f(z) is continuous on a domain D. If any one of the following statement is true, then so are the others:
i. f(z) has an antiderivative F(z) in D;

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ii. the integrals of f(z) along contours lying entirely in D and extending from any fixed point z 1 to any fixed point z 2 all have same value ;
iii.the integral of f(z) around closed contours lying entirely in D all have value zero.

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Corollary: D. in z all for ) ( ) ( D. in f(z) of tive antideriva an is F(z) and D, domain a in continuous is f(z) Let z f z F =
). ( ) ( ) ( Then D. in ENTIRELY lying and , z and z joining contour any is C and D, in points any two be z and z Let 1 2 2 1 2 1 z F z F dz z f C - =

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Example: . ) ( tive antideriva an has ) ( that Note . evaluate to tive antideriva an Use 2 / π z z i i z e z F e z f dz e = =
[ ] π i e e i F i F dz e i i i i z + = - = - = 1 1 ) ( ) 2 / ( 2 / 2 /

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Cauchy - Goursat Theorem: If a function f is analytic at all points interior to and on a simple closed contour C, then 0 ) ( = C dz z f
Example: If C is any simple closed contour, in either direction, then 0 ) exp( 3 = C dz z because the function ) exp( ) ( 3 z z f = is analytic everywhere.

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Defn: A simply connected domain D is a domain such that every simple closed contour within it encloses only points of D.
The set of points interior to a simply closed contour is an example.

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A domain that is not simply connected is said to be multiply connected for example, the annular domain between two concentric circles.
Cauchy – Goursat theorem for a simply connected domain D is as follows:

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## This note was uploaded on 05/14/2010 for the course MATHEMATIC mathe taught by Professor Xyz during the Spring '10 term at Birla Institute of Technology & Science.

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9 cauchy integral formula - ANTIDERIVATIVES Let f(z) be...

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