Mathematical theory of harmonic functions (Δu = 0), their generalisations, and the associated boundary-value problems. Laplace 1782 introduced the Laplace equation and its fundamental solution (Newtonian potential) in celestial mechanics;…
potential-theory
Harmonic function (Δu = 0): mean-value property and max principle
A C² function u: Ω ⊂ R^n → R is harmonic iff Δu = ∂²u/∂x₁² + … + ∂²u/∂x_n² = 0. Three equivalent characterisations: (i) C² with Δu = 0;…
Poisson kernel P_r(θ) reconstructs harmonic u from ∂D data
The Poisson kernel of the unit disk P_r(θ) = (1−r²)/(1 − 2r cos θ + r²) solves the Dirichlet problem for Δu = 0 on D with continuous…
Newtonian potential U_ρ = ∫ Φ(x-y)ρ(y)dy solves ΔU = ρ
Fundamental solution Φ of the Laplace operator: Φ(x) = −(1/(2π)) · log|x| in R², Φ(x) = 1/((n-2)·ω_n · |x|^{n-2}) in R^n for n ≥ 3 (where…
u = x² - y² harmonic: Δu ≡ 0, mean over circle ≡ 0 = u(0)
Exact sympy verification of the mean-value property on the harmonic polynomial u(x, y) = x² − y² (the real part of z² = (x+iy)²). Step 1 —…
Poisson integral normalisation: ∫₀^{2π} P_{1/2}(θ) dθ/(2π) ≡ 1
Sympy-verified symbolic integral pin: for r = 1/2 the normalisation identity (1/(2π)) ∫_0^{2π} P_r(θ) dθ = 1 for the unit-disk Poisson…
Upper-half-plane Green's function G(i, 2i) = log(3)/(2π)
Green's function of the upper half-plane H = {Im(z) > 0} for the Dirichlet Laplacian: G(z, w) = (1/(2π)) · log |z − w̄|/|z − w| —…