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 [1]. 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.

References

[1]	E.T. Whittaker, "An expression of certain known functions as generalized hypergeometric functions" Bull. Amer. Math. Soc. , 10 (1903) pp. 125–134
[2]	E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952)
[3]	H. Bateman (ed.) A. Erdélyi (ed.) , Higher transcendental functions , 1. The gamma function. The hypergeometric functions. Legendre functions , McGraw-Hill (1953)
[4]	A. Kratzer, W. Franz, "Transzendente Funktionen" , Akad. Verlagsgesell. Geest u. Portig K.-D. (1960)
[5]	E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 1. Gewöhnliche Differentialgleichungen , Akad. Verlagsgesell. (1942)