Harmonic Functions and Conjugates: Theory & Examples

A real function u(x, y) is harmonic if it satisfies Laplace's equation u_xx + u_yy = 0. The real and imaginary parts of any analytic function are harmonic: differentiating u_x = v_y and u_y = −v_x gives u_xx + u_yy = 0. A harmonic conjugate v makes u + iv analytic and is found by integrating v_y = u_x, v_x = −u_y.

Laplace's equation describes equilibrium: steady temperature, charge-free electrostatic potential, irrotational flow. The conjugate turns a real equilibrium problem into complex analysis, unlocking its full machinery.

Key Concepts

• Laplacian: Δu = u_xx + u_yy; harmonic means Δu = 0.
• Harmonic conjugate: v with u + iv analytic, unique up to an additive constant.
• Mean value property: u at a point equals its average over surrounding circles.
• Maximum principle: non-constant harmonic functions have no interior maxima or minima.
• Simply connected domain: no holes; there every harmonic function has a conjugate.

Formulas

• ∂²u/∂x² + ∂²u/∂y² = 0.
• Conjugate via dv = −u_ydx + u_xdy.
• For analytic f = u + iv: Δu = Δv = 0, and level curves of u, v form orthogonal families.
• Example: u = x² − y², v = 2xy give f(z) = z².

Worked Example

Show u = x² − y² is harmonic and find its conjugate. u_x = 2x, u_xx = 2, u_y = −2y, u_yy = −2, so u_xx + u_yy = 0. Seek v with v_y = 2x and v_x = 2y. Integrating v_y = 2x in y: v = 2xy + g(x). Then v_x = 2y + g′(x) must equal 2y, so g′ = 0; take g = 0, v = 2xy. Check: u + iv = x² − y² + 2xyi = (x + iy)² = z², analytic, confirming the conjugate.

Common Misconceptions

• "Harmonic means analytic." Harmonic is a real-function property; analytic is a complex-function property. The real part of an analytic function is harmonic, not vice versa.
• "The conjugate is unique." Only up to an additive real constant, like +C in integration.
• "Every harmonic function has a conjugate." On domains with holes it can fail: ln√(x²+y²) is harmonic on the punctured plane but its angle conjugate cannot be made continuous around 0.

Real-World Applications

• Steady heat conduction: plate temperature with fixed boundary values is harmonic.
• Electrostatics: potentials are harmonic; conjugate level curves are field lines.
• Soap films spanning wire frames connect to harmonic conjugates via minimal surfaces.

Harmonic functions are the real shadow of analytic functions, ruled by Laplace's equation and the mean value and maximum principles. The conjugate reassembles the shadow into the full analytic function.