MASTERS COMPREHENSIVE EXAM—COMPLEX VARIABLES—January 2000

1. Indicate whether the given function is analytic on the set where it is defined. No proof

is required.

(i) cos z − 5z
(ii) |z|^{4}
(iii) cos(z̄)
(iv) Arg(z) (the principal value of the argument function)
(v) sin (z^{3} − e^{2z})

2. Let T be the transformation defined by

T (z) = \frac{z + \frac{1}{2}}{1 + \frac{1}{2}z} .

Give a geometric description of the region in the complex plane determined by

\frac{1}{2} < |T (z)| < 1.

3. If C is the following contour from −1 − i to −2, find the contour integral ∫_{C} z dz.

4. Define

f(z) = ∑_{n=1}^{∞} \frac{1}{5^{n}n!z^{n}} + ∑_{n=0}^{∞} \frac{z^{n}}{3n^{2} + n!}.

Given that the series converge for all z ≠ 0, find ∫_{C} f(z) dz if C is the unit circle taken counterclockwise.

5. Use the Residue Theorem to compute with justification the following integral:

∫_{0}^{∞} \frac{cos x}{x^{2} + 1} dx.