Math 103 Final Exam

Problem 1 - What are the Cauchy-Riemann equations and why do they hold for a complex analytic function?

Problem 2 - Let a, b be complex numbers with a ≠ b. Give an explicit branch of log (z-a/z-b).

Problem 3 - Find all Laurent expansions centered at i of the function f(z) = 1/z²-z. For each expansion, indicate the largest open set on which it converges.

Problem 4 - Use the theory of residues to calculate

_-∞∫^∞ dx/z+x⁴.

Problem 5 - Find a conformal mapping of the half-disk, B = {|z| < 1, Im(z) > 0} onto the disk D = {|z| < 1}. Draw a picture indicating the images under your mapping of the arcs of the circles in B which pass through 1 and -1.

Problem 6 - Prove the following theorem. In doing so, you may cite other, weaker, theorems from our text as long as you state them clearly-including all necessary hypotheses.

Theorem: Let D be a bounded domain with piecewise smooth boundary ∂D, and let f(z) be a meromorphic  function on D that extends to be analytic on ∂D, such that f(z) ≠ 0 on ∂D. Then 1/2πi ∫_∂Df′(z)/f(z) dz = N₀ - N_∞, where N₀ is the number of zeros of f(z) in D and N_∞ is the number of poles of f(z) in D, counting multiplicities.

Problem 7 - Let f(z) be defined by the formula

For each of the following statements either gives a careful proof or a careful disproof. (You may cite theorems from our text without proof as long as you state them clearly-including all necessary hypotheses.)

1) There is a sequence of polynomials {p_n(z)}_n≥0 which  converge normally to f on C.

2) There is a sequence of polynomials {p_n(z)}_n≥0 such that on every compact subset, K ⊆ C - R, p_n converges uniformly to f on K.