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Unformatted text preview: ACM95a/100a Lecture Notes Niles A. Pierce Caltech 2008 Cauchys Theorem Theorem 1 If f ( z ) is analytic with f ( z ) continuous on and inside a simple closed contour C then Z C f ( z ) dz = 0 . Proof: Parameterizing C as z ( t ) for a t b and letting f ( z ) = u ( x,y ) + iv ( x,y ), the contour integral can be written in terms of u ( x ( t ) ,y ( t )), v ( x ( t ) ,y ( t )), x ( t ) and y ( t ) as Z C f ( z ) dz = Z b a f ( z ( t )) z ( t ) dt = Z b a ux vy dt + i Z b a vx + uy dt, or equivalently as real line integrals along C Z C f ( z ) dz = Z C u dx v dy + i Z C v dx + u dy. By Greens Theorem, if two real functions P ( x,y ) and Q ( x,y ) are continuous with continuous first order partial derivatives in the region R on and inside a positive simple closed contour C then Z C P dx + Q dy = ZZ R Q x P y dxdy. Now, u ( x,y ) and v ( x,y ) are continuous by the analyticity of f ( z ) and their first order partial derivatives are continuous by the continuity of...
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This note was uploaded on 01/14/2010 for the course ACM 1 taught by Professor Prof during the Fall '09 term at Caltech.
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