View source for Whittaker equation

A linear homogeneous ordinary differential equation of the second order:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097840/w0978401.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

where the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097840/w0978402.png" /> and the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097840/w0978403.png" /> may take arbitrary complex values. Equation (*) represents the reduced form of a degenerate [[Hypergeometric equation|hypergeometric equation]] and was first studied by E.T. Whittaker [[#References|[1]]]. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097840/w0978404.png" /> the Whittaker equation is equivalent to the [[Bessel equation|Bessel equation]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097840/w0978405.png" /> is not an integer, a fundamental system of solutions of the Whittaker equation consists of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097840/w0978406.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097840/w0978407.png" />; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097840/w0978408.png" /> is the Whittaker function (cf. [[Whittaker functions|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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097840/w0978409.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097840/w09784010.png" /> is the Whittaker function.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.T. Whittaker,   "An expression of certain known functions as generalized hypergeometric functions"  ''Bull. Amer. Math. Soc.'' , '''10'''  (1903)  pp. 125–134</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.T. Whittaker,   G.N. Watson,   "A course of modern analysis" , Cambridge Univ. Press  (1952)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Bateman (ed.)  A. Erdélyi (ed.) , ''Higher transcendental functions'' , '''1. The gamma function. The hypergeometric functions. Legendre functions''' , McGraw-Hill  (1953)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A. Kratzer,   W. Franz,   "Transzendente Funktionen" , Akad. Verlagsgesell. Geest u. Portig K.-D.  (1960)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E. Kamke,   "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''1. Gewöhnliche Differentialgleichungen''' , Akad. Verlagsgesell.  (1942)</TD></TR></table>