The Lorentzian function is the singly peaked function given by L(x) = 1/π (1/2 Γ)/((x - x_0)^2 + (1/2 Γ)^2), where x_0 is the center and Γ is a parameter specifying the width. The Lorentzian function is normalized so that integral_(-∞)^∞ L(x) = 1. It has a maximum at x = x_0, where L'(x) = - (16(x - x_0) Γ)/(π[4(x - x_0)^2 + Γ^2]^2) = 0.

Cauchy distribution | damped exponential cosine integral | Gaussian function | hyperbolic secant | logistic distribution | witch of Agnesi