hw3 - Mathematics 185 Intro to Complex Analysis Fall 2009...

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Unformatted text preview: Mathematics 185 Intro to Complex Analysis Fall 2009 M. Christ Solutions to Selecta from Problem Set 3 1.5.15 Let f be an analytic function on D = { z : | z | < 1 } . Suppose that Re( f ( z )) = 3 for all z D . Prove that f is constant. Solution. Write f = u + iv where u,v are real-valued. The hypothesis is that u 3 on D . Therefore the partial derivatives satisfy u x u y 0 on D . By the Cauchy-Riemann equations, v y = v x = 0 at every point of D . Since f = u x + iv x , f ( z ) = 0 for all z D . Since D is connected, this forces f to be constant. Comment. D is convex, and therefore is path-connected; if z,w D then the line segment joining z,w belongs to D . Any path-connected set is connected. 1.5.19(c) Let f ( x ) = z +1 z- 1 . What is the image of the y axis under f ? Solution. Let y R . Let = { z C : | z | = 1 } . f ( iy ) = 1+ iy 1- iy . Note that if w is the numerator 1 + iy , then the denominator 1- iy equals w . Since | w | = | w | for any complex number, both numerator and denominator have the same absolute values. Thus the ratio has absolute value 1, hence is an element of . Thus f ( i R ) . We calculate f ( iy ) = (1 + iy ) 2 (1 + iy )(1- iy ) = 1- y 2 1 + y 2 + i 2 y 1 + y 2 = h ( y ) + ig ( y ) . I claim that as y varies over [0 , 1], g ( y ) = 2 y/ (1 + y 2 ) varies over [0 , 1]; g : [0 , 1] [0 , 1] is a bijection. This is a simple one variable calculus exercise; I leave the details to you. Next, I claim that y 7 f ( iy ) is a bijection from [0 , 1] to the quarter of which lies in the closed first quadrant Q 1 = { x + iy : x 0 and y } . Note that h ( y ) 0 and g ( y ) for y [0 ,...
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This note was uploaded on 10/10/2009 for the course MATH 185 taught by Professor Lim during the Spring '07 term at University of California, Berkeley.

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hw3 - Mathematics 185 Intro to Complex Analysis Fall 2009...

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