Why do the real and imaginary parts of an analytic function both satisfy the Laplace equation?
Think about your answer, then reveal below.
Model answer: An analytic function satisfies the Cauchy-Riemann equations: ∂u/∂x = ∂v/∂y and ∂u/∂y = −∂v/∂x. Differentiating the first equation with respect to x and the second with respect to y and adding gives ∂²u/∂x² + ∂²u/∂y² = 0, i.e., ∇²u = 0. Similarly for v.
This connection is fundamental: complex differentiability imposes the Cauchy-Riemann constraints, and those constraints are precisely the Laplace equation. It explains why complex analysis is so natural for 2D potential theory — the two fields are the same mathematical structure.
Question 2 Short Answer
A conducting wedge has opening angle α. You want to find the potential inside using conformal mapping. What type of mapping would you look for, and why?
Think about your answer, then reveal below.
Model answer: Look for a power-law mapping w = z^(π/α), which maps the wedge sector into a half-plane. The boundary conditions (V = 0 on both sides of the wedge) map to V = 0 on the entire real axis in the w-plane, where the solution is a simple half-plane problem with known form.
Power maps z^n scale angles by n, so choosing n = π/α straightens the wedge into a 180° flat boundary (half-plane). In the half-plane, the solution is V ∝ Im(w) = Im(z^(π/α)), which can be written explicitly in polar coordinates as V ∝ r^(π/α) sin(πθ/α). Near the wedge tip (r → 0), this determines how field strength diverges.