- #1

- 901

- 3

## Homework Statement

Show ln(az) where a is a real number and z = x + iy is harmonic everywhere except z = 0.

## Homework Equations

z = x + iy = rcos(θ) + irsin(θ) = re^iθ

z = u(x,y) + iv(x,y)

Cauchy Riemann test for analyticity:

∂u/∂x = ∂v/∂y

∂u/∂y = -∂v/∂x

## The Attempt at a Solution

ln(az) = ln (rcos(θ) + irsin(θ)) = ln(rcos(θ+2nπ) + irsin(θ+2nπ))

= ln(a*re^iθ) = ln(ar) + i(θ+2nπ) <- this is multivalued, not harmonic.

How do I show it is harmonic?