118Homework4

118Homework4 - z = 2 e it , 0 t (b) the semicircle z = 2 e...

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Homework 4 Due Monday, Sept. 20 1. Let h : R 2 \ { (0 , 0) } → R be given by h ( x, y ) := ln( x 2 + y 2 ). Prove that h is harmonic. Can you ﬁnd a harmonic conjugate for h ? What is the domain of this harmonic conjugate? (Hint: Think of h as the real part of a familiar complex-valued function.) 2. One must be careful when dealing with the “multifunction” log z . For example, prove that the set of values log i 2 is not equal to the set of values 2 log i . 3. Find all z C with the property that cos z = 0. 4. Prove that the “multifunction” f ( z ) = z c is single valued if and only if c is an integer. 5. Use Euler’s Formula e = cos θ + i sin θ to derive trigonometric identities for cos( ) and sin( ), where n N . (Hint: It may help to recall the Binomial Theorem.) 6. Let f ( z ) = z +2 z and evaluate R C f ( z ) dz , where C is the contour (a) the semicircle
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Unformatted text preview: z = 2 e it , 0 t (b) the semicircle z = 2 e it , t 2 (c) the semicircle z = 2 e it , 0 t 2 7. Let f ( z ) = e z and evaluate R C f ( z ) dz , where C is the boundary of the square with vertices at the points 0, 1, 1 + i , and i , the orientation being in the clockwise direction. 8. Prove that if f is the constant function f ( z ) = 1, and if C is any contour from z 1 to z 2 , then R C f ( z ) dz = z 2-z 1 . 9. Let C R be the circle | z | = R (for R > 0) oriented in the counterclockwise direction. Prove that lim R Z C R Log z z 2 dz = 0 . Hint: First show that R C R Log z z 2 dz 2 ( +ln R R ) ....
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This note was uploaded on 02/21/2012 for the course MATH 118 taught by Professor Forde during the Fall '09 term at University of Houston.

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