There are at least two integrals called the Poisson integral. The first is also known as Bessel's second integral, J_n(z) = (1/2 z)^n/(Γ(n + 1/2) Γ(1/2)) integral_0^π cos(z cos θ) sin^(2n) θ d θ, where J_n(z) is a Bessel function of the first kind and Γ(x) is a gamma function. It can be derived from Sonine's integral. With n = 0, the integral becomes Parseval's integral. In complex analysis, let u:U->R be a harmonic function on a neighborhood of the closed disk D^_(0, 1), then for any point z_0 in the open disk D(0, 1), u(z_0) = 1/(2π) integral_0^(2π) u(e^(i ψ))(1 - ( left bracketing bar z_0 right bracketing bar )^2)/( left bracketing bar z_0 - e^(i ψ) right bracketing bar )^2 d ψ.
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