Hypergeometric function

Summary

The Gauss hypergeometric function ${}_2F_1(a, b; c; z)$ is the sum of the hypergeometric series on the unit disc. It is extended by analytic continuation to the cut plane $\C \setminus [1, \infty)$. It solves the hypergeometric differential equation, to which every second-order linear equation with three regular singular points can be reduced. Most classical special functions, such as Legendre functions, complete elliptic integrals, Jacobi polynomials and many elementary functions, are special cases or limits of it.