Math 336 Midterm, April 25, 2008

Name:

One notebook sized page of notes is allowed on the test.

1. Let f(z) be analytic and z(t) = u(t) + iv(t) be differentiable as a function of the real variable t. Let g(t) = f(z(t)). Prove that g′(t) = f′(z(t))z′(t), where f′ is the complex derivative of f and the multiplication of f′ and z′ is multiplication of complex numbers. Carefully explain your reasons. You may quote results about real functions.

2. Let f(z) = u(x, y) + iv(x, y) be real, twice-differentiable on an open set. Suppose that the real and imaginary parts of f(z) and zf(z) are harmonic. Prove that f is analytic.

3. Let f_{n} be a sequence of analytic functions defined on an open set Ω. Suppose f_{n} converges to a function f, uniformly on compact subsets of Ω. Prove that f is analytic and lim_{n→∞} f′_{n} = f′. midterm 4

4. Compute the following contour integral

∫_{|z|=1} \frac{1}{(3z − 1)(2z − 1)}