homework20110919

homework20110919 - | f ( z ) | 2 = u x v y u y v x Hence,...

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Math 417, Fall 2011 Exercise Set #3 Turn in: September 19 1. Does the limit lim z →∞ e z exists? Discuss what happens to the values of e z as z →∞ . 2. Find an analytic function f ( z )= u ( z )+ iv ( z )suchthat u ( x, y )= xy . Is there an analytic function g ( z )whoserea lpartis x 2 y ? Why, or why not? 3. Can 2 x 3 6 xy 2 + x 2 y 2 y be the real part u ( x, y )o fanana ly t i cfun c t ion f ( z )= u ( x, y )+ iv ( x, y )? If so, ±nd all imaginary parts v ( x, y ). 4. Can x 2 y 2 + e y sin x e 6 cos x be the real part u ( x, y )o fanana lyt icfunct ion f ( z )= u ( x, y )+ iv ( x, y )? If so, ±nd all imaginary parts v ( x, y ). 5.
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Unformatted text preview: | f ( z ) | 2 = u x v y u y v x Hence, the absolute value of the derivative of an analytic function f ( z ) is the Jacobian of the dierentiable transformation f ( x, y ) = f ( z ) dened by f ( z ), considered as a function of two variables. 6. Suppose that f ( z ) is analytic in a connected domain C . Show that if f ( z ) is real for all z , then f ( z ) must be a constant function on . 1...
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