Name: Instructor: Bei Hu

Mathematics 425: Complex Variables

Spring Semester 2000

Exam 3 April 14, 2000

This Examination contains 5 problems on 7 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  20
2  20
3  20
4  20
5  20

Total  100

GOOD LUCK 1. (20 points) For each of the following problem, (1) Find all singularities of the following functions in the given domain, (2) Determine the type (removable singularity, pole, essential singularity) of each of the singularities, (3) the order of each of the poles, and (4) the residue at poles. (a) f(z) = \frac{1}{(z^{2} + 4z + 5)^{2}} , z ∈ ℂ. (b) f(z) = \frac{z^{4}}{1 + z^{5}} , Re\{z\} > 0, Im\{z\} > 0.

2 2. (20 points) Find (1) the Laurent-series, and (2) the residue of the following function at z_{0}. (a) f(z) = \frac{e^{z}}{z^{2}} at z_{0} = 0. Answer: (b) f(z) = ze^{1/(z−i)} at z_{0} = i. Answer:

3 3. (20 points) Compute the following real integrals. (a) ∫_{−∞}^{∞} \frac{cos 3x}{x^{2} + 2x + 2}dx Answer: (b) ∫_{−∞}^{∞} \frac{x^{2}}{x^{4} + 1}dx Answer:

4 4. (20 points) (1). Draw a correct path in the complex plane for each of the following problem that will evaluate the real integrals. (2). Find the function f(z) that you will integrate on the path. Do NOT evaluate the integral. (a) ∫_{−∞}^{∞} \frac{x^{2} cos 2x}{x^{4} + 1}dx (b) ∫_{0}^{∞} \frac{\sqrt{x}}{(x^{2} + 2)^{2}} dx (b) ∫_{0}^{∞} \frac{\sqrt{x}}{(x^{2} + 2x + 2)^{2}} dx

5 5. (a). (8 points). State PRECISELY the Rouché’s theorem. (Assumptions and conclusions). (b). (12 points). Find the precise number of zeros of the following polynomial in the given region. p(z) = 2z^{4} − z^{3} + 2z^{2} + 9z + 1 in 1 < |z| < 2. Prove your answer. No credit will be given without a proof.

6 7