Complex Analysis Answers

Suppose z0 is any constant complex number interior to any simple closed

contour C. Show that for a positive integer n,

∮

dz

(z−z0)

n = {

2πi, n = 1

C 0, n > 1.

Express 1/(cos θ − i sin θ) in the form of a + ib and hence prove that

cosθ+isinθ/ cos θ − i sinθ = cos2θ+isin2θ.

Show that | z + w |^2 − | z − w |^2 = 4Re (zŵ).

If z=cosθ+i sinθ,prove that

zn + z−n = 2cos(nθ), z^n − z^−n = 2i sin(nθ).

Apply De Moivre’s formula to express cos 4θ and sin 4θ in terms of cos θ and sin θ.

Determine the distance between 2 − i and 3 + i on the complex plane.

Show that Re (iz) = −Im (z) and Im (iz) = Re (z). Evaluate Re (1/z) and Im (1/z) if

z = x + iy and z 6= 0.

Express 1/(cosθ − i sinθ) in the form of a + ib and hence prove that

cos θ + i sinθ

cos θ − i sinθ = cos 2θ + i sin2θ.

The PDE 2∂

2

z

∂x

2

+2∂

2

z

∂y

2

+4∂

2

z

∂x∂y

=2x

2∂2z∂x2+2∂2z∂y2+4∂2z∂x∂y=2x,

is