Name: Instructor: Bei Hu

Mathematics 425: Complex Variables Spring Semester 2000 Final Examination May 12, 2000

This Examination contains 12 problems on 12 sheets of paper. Do all your work in this booklet and show your computations. Calculators, books, and notes are not allowed.

Scores

Question Possible Actual

1  15
2  10
3  15
4  15
5  10
6  10

Question Possible Actual

7  15
8  15
9  15
10  15
11  10
12  5

Total 150

GOOD LUCK 1. (15 points) Find the limit, if it exists. Give your reason if the limit does not exists. (a) lim_{z→−2} Log z Answer: (b) lim_{n→∞}(−i)^{n} Answer:

2. (10 points) Find all 3 roots of z^{3} = 27i and show them in the picture. [figure: 1; i] 3. (15 points) (a) If f = u + iv is analytic, write the Cauchy-Riemann equation for u and v. (b) Is the function f(z) = x^{2} + y^{2} − i2xy (z = x + iy) analytic? Justify your answer (no credit will be given if reason is not given). (c) Is the function g(z) = x^{2} − y^{2} + i2xy (z = x + iy) analytic? Justify your answer (no credit will be given if reason is not given).

3 4. (15 points) Calculate all values of the following expressions and present the results in standard form a + bi. Simplify your answer. (a) i^{1/2} Answer: (b) log(−4) Answer: (c) cos(−i) Answer:

4 5. (10 points) Find the radius of convergence of the following power series. (a) ∑_{n=0}^{∞} \frac{(2n)^{3}(2n + 3)^{4}}{3^{n}} z^{2n} , Answer: (b) ∑_{n=0}^{∞} \frac{(3 − 4i)^{n}}{5n} (z + 2i)^{n} , Answer:

5 6. (10 points) Find the residue of the following functions for all the poles in the given region. (a) f(z) = \frac{z^{5}}{z^{6} + 1} Re\{z\} > 0, Im\{z\} > 0. (b) g(z) = \frac{3z^{2}}{(z^{2} − 4z + 5)^{2}} Re\{z\} > 0, Im\{z\} > 0.

6 7. (15 points) Compute the following integrals (a) ∫_{|z−5i|=4} \frac{e^{z/2}}{z^{2} + 4}dz. Answer: (b) ∫_{|z|=π} \frac{z^{2} + 2z + 2}{e^{z} − 1}dz Answer:

7 8. (15 points) Compute the following integrals. (a) ∫_{−∞}^{∞} \frac{1}{1 + x^{4}} dx. Answer: (b) ∫_{−∞}^{∞} \frac{cos 2x}{x^{2} − 2x + 2}dx Answer:

8 9. (a) (5 points). State PRECISELY the Maximum Modulus Theorem (for the modulus of a complex function). (b) (10 points). Suppose that f is analytic on a domain D which contains a simple closed curve γ. Suppose that (i). |f(z)| is a constant on γ and (ii). f(z) has no zeros in D. Prove that f(z) is a constant. (Remark: the assumption (ii) is necessary. The function f(z) = z is not a constant in the interior of the unit disk, but |f(z)| = 1 on the boundary of the unit disk).

9 10. (a) (5 points). State PRECISELY the Riemann Mapping Theorem. (b) (10 points). Find a conformal mapping which maps the domain D = {z | π/4 < argz < 3π/4} (one-to-one and onto) to the unit disk.

10 11. (10 points). Determine whether you can use the Riemann Mapping Theorem for the following domains, and JUSTIFY your answer. (a) D = {z | |z| < 2, Im{z} > 0}. (b) D = {z | 1 < |z| < 2}.

11 12. (5 points). For the function f(z) = z^{2}, draw the level curves of the real and imaginary parts of f(z). Find the angle at the intersection of the level curves.

12