MA 525 FINAL-2 PRACTICE April, 2000

Name:

1. Find (with any method) conformal maps of the following regions onto the upper halfplane.

2. How many roots do these equations have in the specified regions?

a) z^{4} − 3z + 1 = 0 in |z| < 1,

b) z = 2 − e^{−z} in the right half-plane; on the positive ray.

c) z^{6} − 6z + 10 in |z| > 1.

3. Evaluate the integrals

∫_{−∞}^{∞} \frac{e^{ix}}{x^{2} − 2ix − 2}dx, ∫_{0}^{∞} \frac{Logx}{(x + 1)^{2}} dx, ∫_{0}^{∞} \frac{1 − cos 2x}{x^{2}}dx.

4. Find the sum of the series

∑_{n=1}^{∞} (−1)^{n}n^{3},

by integrating z^{−3} csc πz over an appropriate countur.

5. Find the Laurent series expansions:

a) of 1/(z − 2) in 1 < |z − 3| < ∞,

b) of z^{2} sin π \frac{z+1}{z} in 0 < |z| < ∞.

7. Find and classify all isolated singularities of these functions:

\frac{sin πz^{2}}{sin πz} , exp tan z, \frac{1}{e^{z} + 1}.

8. Find the radius of convergence of these power series

\frac{Logz}{z − 1} = ∑_{n=0}^{∞} a_{n}(z − 3)^{n},

\frac{Logz}{z − 1} = ∑_{n=0}^{∞} b_{n}(z + 1 + i)^{n},

∑_{n=0}^{∞} 2^{n}z^{n!}.

9. Expand into Laurent series:

exp(1/(z − 1)) in |z| > 1,

Log(1/(1 − z))in |z| > 1,

(z^{2} − 1)^{−1}in |z| > 1,

(z^{2} − 1)^{−1}in 0 < |z + 1| < 2,

z^{2} sin(1/z)in |z| > 0.

Find a harmonic conjugate to

e^{x}(x cos y − y sin y).

10. Yes or no: a) Log|z| is harmonic in 0 < |z| < ∞, b) Log|z| is the real part of some analytic function in 0 < |z| < ∞,

c) ∫ e^{1/z^{2}}dz is path independent in 0 < |z| < ∞,

d) ∫ ze^{1/z^{2}}dz is path independent in 0 < |z| < ∞,

e) | cos z| for |z| ≤ 1 has maximal value at the point 0.

f) if P is a non-constant polynomial, whose all zeros belong to the unit disc, then ∫_{|z|=2}(1/P (z))dz = 0.