The Hilbert transform (and its inverse) are the integral transform g(y) | = | ℋ[f(x)] = 1/π P V integral_(-∞)^∞ (f(x) d x)/(x - y) f(x) | = | ℋ^(-1)[g(y)] = - 1/π P V integral_(-∞)^∞ (g(y) d y)/(y - x), where the Cauchy principal value is taken in each of the integrals. The Hilbert transform is an improper integral. In the following table, Π(x) is the rectangle function, sinc(x) is the sinc function, δ(x) is the delta function, I I(x) and I I(x) are impulse symbols, and _1 F_1(a;b;x) is a confluent hypergeometric function of the first kind.

Abel transform | Fourier transform | improper integral | integral transform | Titchmarsh theorem | Wiener-Lee transform