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Unformatted text preview: f is constant. ± 19. Let G be a region and deﬁne G * = { z : ¯ z ∈ G } If f : G → C is analytic, prove that f * : G * → C , deﬁned by f * ( z ) = f (¯ z ), is also analytic. We let f = u ( x, y ) + iv ( x, y ) where u and v are realvalued functions and note that f * = u * ( x,y )iv * ( x,y ). Since f is analytic it must satisfy the Cauchy Riemann equations: 1 u x = v y u y =v x We note that since f * has a domain in G * , both u * and v * ultimately lose the minus sign on the y . So we ﬁnd, ∂ ∂x u * ( x,y ) = u * x ( x,y ) = u * x ( x, y ) = u x ( x, y ) Similarly we ﬁnd the equations u * y ( x,y ) =u y ( x, y ) v * x ( x,y ) =v x ( x, y ) v * y ( x,y ) = v y ( x, y ) which give u * x = v * y u * y =v * x since the necessary continuity conditions are satisﬁed by the fact that f is analytic, we have obtained the desired result. 2...
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This note was uploaded on 10/20/2009 for the course MATH 814 taught by Professor Cong during the Three '09 term at University of Adelaide.
 Three '09
 Cong
 Derivative

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