Complex Analysis - Maths KU

426 Chapter 7 Conformal Mappings
y
∇ φ = 0 θ
2
φ = | θ|
Figure 7.39 Figure for Example 3
x
EXAMPLE 3 Using the Poisson Integral Formula
Use the Poisson integral formula (12) to find a solution of the Dirichlet
problem
Solve:
∂2φ ∂x2 + ∂2φ ∂y2 =0, x2 + y2 < 1
Subject to: φ(cos θ, sin θ) =|θ|, −π 0.
1. 2.
y
∇ φ = 0
2
–1 0 1
φ = 0 φ=
–1 φ = 1 φ = 0
Figure 7.40 Figure for Problem 1
x
∇ φ = 0
2
–2 0 1
x
φ = –1 φ = 5 φ = 1 φ = 0
y
Figure 7.41 Figure for Problem 2