417, Math Fall 2011
Exercise Set #3
Turn in: September 19
1. Does the limit lim ez exists? Discuss what happens to the values of ez as z .
z
2. Find an analytic function f (z) = u(z) + i v(z) such that u(x, y) = xy. Is there an analytic function g(z) whose real part is x2 y? Why, or why not? 3. Can 2x3 - 6xy 2 + x2 - y 2 - y be the real part u(x, y) of an analytic function f (z) = u(x, y) + i v(x, y)? If so, find all imaginary parts v(x, y). 4. Can x2 - y 2 + e-y sin x - e6 cos x be the real part u(x, y) of an analytic function (z) f = u(x, y) + i v(x, y)? If so, find all imaginary parts v(x, y). 5. Let f (z) = u(x, y) + iv(x, y) be an analytic function. Show that u v u v - x y y x Hence, the absolute value of the derivative of an analytic function f (z) is the Jacobian of the differentiable transformation f (x, y) = f (z) defined by f (z), considered as a function of two variables. |f (z)| =
2
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 .
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