Show u = ½log(x²+y²) harmonic; conjugate

Harmonic function

A real function u(x, y) is harmonic in a region if it has continuous second partial derivatives and satisfies Laplace's equation u_xx + u_yy = 0 there.

Show that u = ½ log(x² + y²) is harmonic

u_x  = x/(x² + y²)
u_y  = y/(x² + y²)
u_xx = [(x² + y²) − 2x²]/(x² + y²)² = (y² − x²)/(x² + y²)²
u_yy = [(x² + y²) − 2y²]/(x² + y²)² = (x² − y²)/(x² + y²)²

u_xx + u_yy = 0 for (x, y) ≠ (0, 0). Hence u is harmonic.

Harmonic conjugate v

By the C–R equations v_x = −u_y and v_y = u_x.

v_y = x/(x² + y²)
Integrate w.r.t. y:   v = tan⁻¹(y/x) + g(x)
Differentiate w.r.t. x: v_x = −y/(x² + y²) + g′(x)

This must equal −u_y = −y/(x² + y²), so g′(x) = 0 and g is a constant.

Answer: v = tan⁻¹(y/x) + c (that is, the argument θ).

The analytic function is f(z) = u + iv = ln r + iθ = log z.