View source for Confluent hypergeometric equation

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''degenerate hypergeometric equation''

The second-order ordinary linear differential equation

$$ \tag{1 }
zw  ^ {\prime\prime} + ( \gamma - z) w  ^  \prime  - \alpha w  =  0,\ \ 
\alpha , \gamma = \textrm{ const } ,
$$

or, in self-adjoint form,

$$ 
( e ^ {- z } z  ^  \gamma  w  ^  \prime  )  ^  \prime  -
\alpha e ^ {- z } z ^ {\gamma - 1 } w  =  0.
$$

In general, the variables  $  z, w $
and the parameters  $  \alpha , \gamma $
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  $  x = 0 $
is a regular singular point for equation (1), while the point  $  x = \infty $
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]]  $  \Phi ( \alpha ;  \gamma ;  z) $. 
If  $  \gamma $
is not an integer, the general solution of equation (1) may be written in the form

$$ \tag{2 }
w  =  C _ {1} \Phi ( \alpha ;  \gamma ;  z) +
C _ {2} z ^ {1 - \gamma } \Phi ( \alpha + 1 - \gamma ;  2 - \gamma ;  z),
$$

where  $  C _ {1} $
and  $  C _ {2} $
are arbitrary constants. This representation is valid in the complex  $  z $-
plane with the slit  $  ( - \infty , 0) $. 
If  $  \gamma $
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  $  z $-
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

$$ 
( a _ {0} z + b _ {0} ) w  ^ {\prime\prime} +
( a _ {1} z + b _ {1} ) w  ^  \prime  +
( a _ {2} z + b _ {2} ) w  =  0,
$$

$$ 
a _ {i} , b _ {i}  =  \textrm{ const } ,
$$

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 $1+\alpha\beta x/ (\gamma\cdot 1)+\cdots$"  ''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>