MIDTERM COMPLEX ANALYSIS No credit will be given for an answer without reasoning. 1. Let u(x, y) = y

MIDTERM COMPLEX ANALYSIS

No credit will be given for an answer without reasoning.

1. Letu(x, y) = y³− 3x²y.

(1) [5%] Show thatu is harmonic.

(2) [5%] Determine a harmonic conjugate ofu.

2.

(1) [5%] Letf (z) = (z − 1)/(z + 1). Is f conformal at z = 0?

(2) [5%] Show thatcos(z + 2π) = cos z.

3.

(1) [5%] Calculate(√

3 + i)³(1 −√ 3i)². (2) [5%] Find all values of(−1)ⁱ.

4.

(1) [5%] Evaluate

I

|z|=2

e^z z²− 9dz.

(2) [5%] Evaluate

I

|z−2|=2

e^z z²− 9dz.

5.

(1) [5%] Give a parametrization of the following contour starting fromi to 2 + i with parameter 0 ≤ t ≤ 1.

-4 -3 -2 -1 0 1 2 3 4

-2 -1 0 1 2

(2) [5%] EvaluateR

γf where f (x + iy) = x²+ iy² andγ is the line joining 1 to i.

6. Letf : C → C be analytic. Define g : C → C by g(z) = [f(¯z)]². Show thatg is analytic.

7. Find the maximum of|e^iz| on the set |z| ≤ 2.

1

2 MIDTERM COMPLEX ANALYSIS

8. [10%] We define

∂

∂ ¯z = 1 2

· ∂

∂x+ i ∂

∂y

¸ . Check that iff = u + iv is an analytic function, then ∂f /∂ ¯z = 0.

9. [10%] Suppose thatf (z) is an entire function such that f (z)/zⁿ is bounded on the entire complex plane. What can you say aboutf ?

10. [10%] Letf be analytic on D : |z| < 1 and continuous on ¯D : |z| ≤ 1. Show that if f is nonzero on D, then |f(z)| attains its minimum value on the boundary of D.