Name: Instructor: Bei Hu

Mathematics 425: Complex Variables

Spring Semester 2000

Exam 2 March 27, 2000

This Examination contains 7 problems. Do all your work in this booklet and show your computations. Calculators, books, and notes are not allowed.

Scores

Question Possible Actual

1  10
2  10
3  15
4  15
5  10
6  15
7  25

Total  100

GOOD LUCK 1. (10 points) Without evaluating, find the radius of convergence of the power series (centered at z_{0}) of the following functions. (a) f(z) = \frac{e^{z}}{z^{2} − 4}, z_{0} = 0. Answer: (b) g(z) = \frac{e^{z}}{z^{2} − 4}, z_{0} = 3. Answer: (c) h(z) = \frac{e^{z}}{z^{2} − 4}, z_{0} = i. Answer:

2 2. (10 points) (a) State PRECISELY Liouville’s theorem. (b) Suppose that f is an entire function and Re(f) ≤ C for all z. Prove that f is a constant.

3 3. (15 points) Find the radius of convergence of the following power series. (a) ∑_{n=0}^{∞}(n + 1)(n + 3)(z − 2 − i)^{n} Answer: (b) ∑_{n=0}^{∞} \frac{i^{n}n^{2}}{5^{n}} (z + 2)^{n} Answer: (c) ∑_{n=0}^{∞} \frac{4n}{3^{n}}z^{2n} Answer:

4 4. (15 points) Find the power series about the origin for the given function. (a) \frac{2 + z}{2 − z} (b) z^{4}e^{−2z}

5 5. (10 points) True or False questions. Determine whether the following statement is true or false. (a) If a function f(z) has a continuous first order derivative on a domain D, then derivatives of f(z) of any orders exist. (b) If a function f(z) has a power series expansion in the disc \{|z − z_{0}| < R\}, then f(z) is an analytic function in this disc \{|z − z_{0}| < R\}. (c) For an analytic function f on a domain, if Re(f) is a constant, then Im(f) is also a constant. (d) For any analytic function f on a domain D, and a smooth (but not closed) curve inside D γ : z = γ(t), a ≤ t ≤ b, γ(a) ≠ γ(b), we have ∫_{γ} f(z)dz = 0. (e) If f(z) is an analytic function on a domain, then its conjugate \overline{f(z)} is also an analytic function.

6 6. (15 points) For each of the following functions u(x, y) decide whether it can be the real part of an analytic function f. If so, find v(x, y) so that f = u + iv is analytic. (a) u(x, y) = 3x^{2}y^{2} Answer: (b) u(x, y) = 2xy − x^{2} + y^{2} Answer:

7 7. (25 points) Let |z − a| = r denote the circle with radius r and center a oriented counterclockwise. Find the values of the following integrals. Indicate which theorem you used to obtain your answer. (a) ∫_{|z|=4} \frac{1}{z^{2} + 4}dz Answer: (b) ∫_{|z|=5} \frac{e^{3z}}{(z − 3)(z + 7)^{2}} dz Answer:

8

(c) ∫_{|z−i|=20} \frac{e^{2z}}{z^{2} + 3z + 2}dz

Answer:

(d) ∫_{0}^{2π} \frac{dθ}{2 + cos θ}

Answer:

10