(2) [5%] Evaluate Z Γ ¯ z dz where Γ is the right half of the circle |z

FINAL OF COMPLEX ANALYSIS

No credit will be given for an answer without reasoning.

If f is analytic inside and on the simple closed positively oriented contour Γ and if z0 is inside Γ. The (generalized) Cauchy integral formula:

f⁽ⁿ⁾(z₀) = n!

2πi Z

Γ

f (z) (z − z₀)ⁿ⁺¹dz.

1.

(1) [5%] Let f (z) be the M¨obius transformation that maps i to 0, 1 + i to 1 and 2 − i to ∞. Find f (0).

(2) [5%] Find the cubic roots of i.

2.

(1) [5%] Write the number sin(2i) in the form a + bi (a, b ∈ R).

(2) [5%] Evaluate Z

Γ

¯ z dz

where Γ is the right half of the circle |z| = 3 from −3i to 3i.

3.

(1) [5%] Find and classify the isolated singularities of the function tan z.

(2) [5%] Evaluate I

|z|=2π

tan z dz

(where the circle traverses once in the positive direction) by means of the Cauchy residue theo- rem.

4.

(1) [5%] Let f be analytic inside and on the simple closed contour Γ. What is the value of 1

2πi Z

Γ

f (z) (z − z₀)² dz

when z0lies outside Γ?

(2) [5%] Let C be the ellipse ^x₄² + ^y₉² = 1 (on the complex plane) traversed once in the positive direction, and define

G(z) :=

Z

C

ζ²− ζ + 2 ζ − z dζ (for z insider C). Find G⁰(i).

1

2 FINAL OF COMPLEX ANALYSIS

5.

(1) [5%] Use Rouch´e theorem to show that the polynomial z⁶+ 5z²− 1 has exactly two zeros in the disk |z| < 1.

(2) [5%] Is e^1/za meromorphic function on the whole complex plane? Why or why not?

6.

(1) [5%] Find a M¨obius transformation which maps the interior of the circle |z − 1| = 1 onto the upper half-plane Im(z) > 0.

(2) [5%] Let D^∗be the domain consisting of all points of the complex plane except those lying on the non-positive real axis. Is there any analytic function that maps D^∗ onto the whole complex plane? Why or why not?

7.

(1) [5%] Let Γ₁ be the circle on the xy-plane centered at (1, 1) with radius 1; and Γ₂be the circle centered at (1, 1) with radius 2. Find a harmonic function φ(x, y) such that φ = 0 on Γ1 and φ = 20 on Γ₂.

(2) [5%] Let f (z) =P_∞

k=0(k³/3^k)z^k. Compute I

|z|=1

f (z) z² dz

where the circle |z| = 1 traversed once in the positive direction.

8.

(1) [5%] Classify the behavior at ∞ for the function cos z/z²(if it is a zero or a pole, give its order).

(2) [5%] Let bC := C ∪ {∞} denote the extended complex plane. Suppose that f is an analytic function on bC. Show that f is a constant.

9. Check that Z _π/2

0

1

1 + sin²θdθ =

√2

4 π.

10. Let m be a positive integer. Show that Z _∞

0

sin mx

x dx = π 2.