Name: Instructor: Bei Hu

Mathematics 425: Complex Variables

Spring Semester 2000

Exam 1 February 18, 2000

This Examination contains 8 problems on 7 sheets of paper including the front cover and one extra sheet on the back. 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  15
6  10
7  10
8  15

Total  100

GOOD LUCK 1. (10 points) (a) For z = 1 + 2i, w = 2 + i, compute \overline{z}w + 2z\overline{w} and put your answer in the form of a + bi. Answer: (b) Compute (\frac{1 − i}{\sqrt{2}})^{80}. Answer:

2. (10 points) Let α be a complex number such that |α| < 1. Show that the set of all z with |z − α| = |1 − \overline{α}z| is the circle \{z : |z| = 1\}.

3. (15 points) Find all 3 roots of z^{3} = −5 − 5i. Answer:

4. (15 points) Find the limit, or explain why it does not exist. (a) f(z) = \frac{z^{3} + i}{z − i}, z_{0} = i. Find lim_{z→z_{0}} f(z). Answer: (b) Find lim_{n→∞} (\frac{4 + 3i}{5})^{n}. Answer:

5. [figure: (15 points) Is the set described in the following open? convex? Is the interior of the set; connected? Show the set and sketch your ideas for the answers on the diagram.; (a) {z = x + iy, | − 4 < x < 4, −2 < y < x/4}.; 1; i; Answer:; (b) {z = re^{iθ} | 1 < r < 4, −π/4 ≤ θ ≤ 3π/4}.; 1; i; Answer:] 4 6. (10 points) Compute the value(s) of the following expressions, and write in the format a + bi. (a) (1 + i)^{1+i} Answer: (b) log(−1) Answer:

7. (10 points) Determine whether the following series converge, indicating the test you used. (a) ∑_{n=1}^{∞} \frac{n i^{n}}{2^{n}} Answer: (b) ∑_{n=1}^{∞} \frac{n!(\sqrt{3} + i)^{n}}{n^{n}} Answer:

8. (15 points) Compute the integral ∫_{γ} \frac{1}{z^{2}} dz, where γ is the semicircle from 2i to −2i, passing through −2. 6 7