# Whittaker equation

A linear homogeneous ordinary differential equation of the second order:

$$w''+\left(\frac{1/4-\mu^2}{z^2}+\frac\lambda z-\frac14\right)w=0,\label{*}\tag{*}$$

where the variables $z,w$ and the parameters $\lambda,\mu$ may take arbitrary complex values. Equation \eqref{*} represents the reduced form of a degenerate hypergeometric equation and was first studied by E.T. Whittaker . For $\lambda=0$ the Whittaker equation is equivalent to the Bessel equation. If $2\mu$ is not an integer, a fundamental system of solutions of the Whittaker equation consists of the functions $M_{\lambda,\mu}(z)$ and $M_{\lambda,-\mu}(z)$; here $M_{\lambda,\mu}(z)$ is the Whittaker function (cf. Whittaker functions). For any value of the parameters the general solution of the Whittaker equation may be written in the form of a linear combination

$$w=C_1W_{\lambda,\mu}(z)+C_2W_{-\lambda,\mu}(-z),$$

where $W_{\lambda,\mu}(z)$ is the Whittaker function.

How to Cite This Entry:
Whittaker equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Whittaker_equation&oldid=44683
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article