MIDTERM 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 z

MIDTERM COMPLEX ANALYSIS

No credit will be given for an answer without reasoning.

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

f⁽ⁿ⁾(z₀) = n!

2πi Z

Γ

f(z) (z − z0)ⁿ⁺¹dz.

1.

(1) [5%] Computearg i.

(2) [5%] Let x, y be real numbers. Show that|e^x+iy| = e^x. 2.

(1) [5%] Calculate(1 −√ 3i)³(√

3 + i)².

(2) [5%] Represent the number _(1−i)¹ ₂ in polar form.

3. Let f(z) = z⁴− 4z³+ 6z²− 4z + 5.

(1) [5%] Check that i is a root of f(z) = 0.

(2) [5%] Find all other roots.

4.

(1) [5%] Find all values of z such that e^z = −4.

(2) [5%] Find the principal value of(1 + i)^πi. 5.

(1) [5%] Let D be the rectangle T in z plane with vertices 1, 1 + πi, −1 + πi, −1. Describe the image of T in w plane under the map w= e^z.

(2) [5%] Show that if v is a harmonic conjugate of u in a domain D, then uv is harmonic in D.

6.

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

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

-2 -1 0 1 2

1

2 MIDTERM COMPLEX ANALYSIS

(2) [5%] Compute the line integral

Z

C

¯ z dz where C is the line segment from−1 − i to 3 + i.

(3) [5%] Compute

Z

C

z+ i z³+ 2z² dz,

where C is the circle|z + 2 − i| = 2 traverse once counterclockwise.

7. The Legendre polynomial Pn(z) is defined by Pn(z) = 1 2ⁿn!

dⁿ dzⁿ

h

(z²− 1)ⁿi . (1) [5%] Use Cauchy’s formula to show that

Pn(z) = 1 2πi

Z

Γ

(ζ²− 1)ⁿ 2ⁿ(ζ − z)ⁿ⁺¹ dζ.

whereΓ is simple closed positively oriented contour containing z in its interior.

(2) [5%] Taking C to be the circle with center at z and radiusp|z²− 1|, show that P_n(z) = 1

π Z π

0

(z +p

z²− 1 cos θ)ⁿdθ, which is known as Laplace’s formula.

(3) [5%] Using Laplace’s formula, show that P2(z) = ¹₂(3z²− 1).

8. [10%] Let f be analytic in the simply connected domain D and let z₁and z₂be two complex numbers that lie interior to the simple closed contourΓ having positive orientation that lies in D. Show that

f(z₂) − f(z1) z₂− z1

= 1

2πi Z

Γ

f(z)

(z − z1)(z − z2)dz.

State what happen when z₂ → z1.

9. [10%] Let f be a non-constant analytic function in the closed disk|z| ≤ 1. Suppose that |f(z)| is constant on the circle|z| = 1. Show that f has a zero in the domain |z| < 1.