MASTERS COMPREHENSIVE EXAM—COMPLEX VARIABLES—January 1999

1. Use the definition of the derivative to prove that f′(z) = −\frac{1}{z^{2}} if f(z) = \frac{1}{z} (z ≠ 0).

2. Define f(z) = \frac{sin z}{z} if z ≠ 0 , and define f(0) = 1. Prove that f is an entire function.

3. Let C be the unit circle taken counterclockwise, and define

g(w) = ∫_{C} \frac{2z^{2} − z − 2}{z − w} dz

for |w| ≠ 1.

(a) Compute g(\frac{1}{2}).

(b) Compute g(2).

4. Let f(z) = e^{1/z}, and let C be the unit circle taken counterclockwise.

(a) Determine the Laurent series of f(z) at z = 0.

(b) Determine the residue of f(z) at z = 0.

(c) Is z = 0 a removable singular point, a pole, or an essential singular point of f(z)?

(d) Evaluate the contour integral of z^{2}f(z) along C.

5. Let a > 1. Using residue theory, prove that

∫_{0}^{π} \frac{dθ}{a + cos θ} = \frac{π}{\sqrt{a^{2} − 1}}.