- #1
chill_factor
- 903
- 5
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?