The Laplace Equation

Complex Analysis · Axiom Academy

Why the real and imaginary parts of every analytic function are automatically harmonic — and why that makes complex analysis a physics tool. 1. What "Harmonic" Really Means A function is harmonic when it satisfies Laplace's equation. In two dimensions that reads: The curvatures in the two directions exactly cancel That cancellation has a beautiful geometric meaning: at every point, the value of a harmonic function equals the average of its values on any circle drawn around that point. No local bumps, no local pits — the surface is pulled perfectly taut, like a soap film. Watch a probe circle ride over the harmonic surface u = x^2 - y^2 : the running average of the boundary (orange) lands exactly on the center value (blue) at every spot. 2. Analyticity Forces Harmonicity Here is the bridge to complex analysis. If f(z) = u + iv is analytic , its real and imaginary parts obey the Cauchy–Riemann equations . Differentiate them once more and the two halves of the Laplacian fall straight out: The animation makes the cancellation visible: the two second-derivative terms v_ yx and v_ xy slide together and annihilate, leaving u_ xx + u_ yy = 0 . Here u = x^2 - y^2 and v = 2xy . Check u : u_ xx = 2 and u_ yy = -2 , so u_ xx + u_ yy = 0 . Check v : v_ xx = 0 and v_ yy = 0 , sum 0 . Both parts are harmonic — exactly as the proof promises. The same identical argument works for v because it satisfies its own Cauchy–Riemann pair. 3. The Two Halves Are a Physics Pair

This is the written version of the interactive lesson above. See the full Complex Analysis course.