View source for Confluent hypergeometric equation

''degenerate hypergeometric equation''

The second-order ordinary linear differential equation

<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/c/c024/c024690/c0246901.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

or, in self-adjoint form,

<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/c/c024/c024690/c0246902.png" /></td> </tr></table>

In general, the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024690/c0246903.png" /> and the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024690/c0246904.png" /> may assume complex values. The [[Whittaker equation|Whittaker equation]] is a reduced form of equation (1). Equation (1) is closely connected with the [[Hypergeometric equation|hypergeometric equation]]. The confluent hypergeometric equation can be regarded as an equation obtained from the [[Riemann differential equation|Riemann differential equation]] as a result of the merging of two singular points. The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024690/c0246905.png" /> is a regular singular point for equation (1), while the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024690/c0246906.png" /> is a strong singular point (cf. [[Singular point|Singular point]]). E.E. Kummer [[#References|[1]]] was the first to undertake a systematic study of the solutions of equation (1).

The solutions of equation (1) are expressed through the [[Confluent hypergeometric function|confluent hypergeometric function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024690/c0246907.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024690/c0246908.png" /> is not an integer, the general solution of equation (1) may be written in the form

<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/c/c024/c024690/c0246909.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024690/c02469010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024690/c02469011.png" /> are arbitrary constants. This representation is valid in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024690/c02469012.png" />-plane with the slit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024690/c02469013.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024690/c02469014.png" /> is an integer, the general solution has a more complicated form (may contain logarithmic terms). Functions other than those in (2) (e.g. [[Whittaker functions|Whittaker functions]] [[#References|[2]]], [[#References|[3]]]) can also be selected as a fundamental system of solutions of equation (1). Solutions of (1) can also be represented by contour integrals in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024690/c02469015.png" />-plane.

Many second-order ordinary linear differential equations (e.g. the [[Bessel equation|Bessel equation]]) can be reduced to equation (1) by a transformation of the unknown function and of the independent variable [[#References|[4]]]. In particular, equations of the type

<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/c/c024/c024690/c02469016.png" /></td> </tr></table>

<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/c/c024/c024690/c02469017.png" /></td> </tr></table>

can be integrated using the confluent hypergeometric function.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.E. Kummer,   "Ueber die hypergeometrische Reihe <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024690/c02469018.png" />"  ''J. Reine Angew. Math.'' , '''15'''  (1836)  pp. 39–83; 127–172</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Krazer,   W. Franz,   "Transzendente Funktionen" , Akademie Verlag  (1960)</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">  E. Kamke,   "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''1. Gewöhnliche Differentialgleichungen''' , Chelsea, reprint  (1971)</TD></TR></table>