Home_4 - f is constant ± 19 Let G be a region and deﬁne...

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Homework 4 Stephen Taylor June 4, 2005 Page 43 - 44 1. Show that f ( z ) = | z | 2 = x 2 + y 2 has a derivative only at the origin. We first note a theorem from Complex Variables and Applications by Brown: Theorem 1. Let the function f ( z ) = u ( x, y ) + iv ( x, y ) be defined throughout some neighborhood of the point z 0 = x 0 + iy 0 , and suppose that the first-order partial derivatives of the functions u and v with respect to x and y exist everywhere in that neighborhood. If those partial derivatives are continuous at ( x 0 , y 0 ) and satisfy the Cauchy-Riemann equations at ( x 0 , y 0 ) , then f ( z 0 ) exists. If the Cauchy-Riemann equations are to hold at a point ( x, y ), then 2 x = 0 and 2 y = 0. So x = y = 0 Since the partial derivatives satisfy the necessary continuity requirements of the theorem, we find that f ( z ) exists only for z = 0 as desired. 14. Suppose f : G C is analytic and that G is connected. Show that if f ( z ) is real for all z in G then f is constant. Since f ( z ) is real for all z it has a representation f ( z ) = u ( z ) where u is a real-valued function. Since f must satisfy the Cauchy-Riemann equations, we find u x = u y = 0. So u must be constant which implies

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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 real-valued 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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