Namespaces
Variants
Actions

Difference between revisions of "Confluent hypergeometric equation"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
 +
<!--
 +
c0246901.png
 +
$#A+1 = 17 n = 1
 +
$#C+1 = 17 : ~/encyclopedia/old_files/data/C024/C.0204690 Confluent hypergeometric equation,
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
''degenerate hypergeometric equation''
 
''degenerate hypergeometric equation''
  
 
The second-order ordinary linear differential 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>
+
$$ \tag{1 }
 +
zw  ^ {\prime\prime} + ( \gamma - z) w  ^  \prime  - \alpha w  = 0,\ \
 +
\alpha , \gamma = \textrm{ const } ,
 +
$$
  
 
or, in self-adjoint form,
 
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>
+
$$
 +
( e ^ {- z } z  ^  \gamma  w  ^  \prime  )  ^  \prime  -
 +
\alpha e ^ {- z } z ^ {\gamma - 1 } w  = 0.
 +
$$
  
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).
+
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]] <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
+
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
  
<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>
+
$$ \tag{2 }
 +
= C _ {1} \Phi ( \alpha ; \gamma ; z) +
 +
C _ {2} z ^ {1 - \gamma } \Phi ( \alpha + 1 - \gamma ; 2 - \gamma ; z),
 +
$$
  
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.
+
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
 
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>
+
$$
 +
( a _ {0} z + b _ {0} ) w  ^ {\prime\prime} +
 +
( a _ {1} z + b _ {1} ) w  ^  \prime  +
 +
( a _ {2} z + b _ {2} ) w  = 0,
 +
$$
  
<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>
+
$$
 +
a _ {i} , b _ {i}  = \textrm{ const } ,
 +
$$
  
 
can be integrated using the confluent hypergeometric function.
 
can be integrated using the confluent hypergeometric function.

Revision as of 17:46, 4 June 2020


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 is a reduced form of equation (1). Equation (1) is closely connected with the hypergeometric equation. The confluent hypergeometric equation can be regarded as an equation obtained from the 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). E.E. Kummer [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 $ \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 [2], [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) can be reduced to equation (1) by a transformation of the unknown function and of the independent variable [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

[1] E.E. Kummer, "Ueber die hypergeometrische Reihe " J. Reine Angew. Math. , 15 (1836) pp. 39–83; 127–172
[2] A. Krazer, W. Franz, "Transzendente Funktionen" , Akademie Verlag (1960)
[3] H. Bateman (ed.) A. Erdélyi (ed.) , Higher transcendental functions , 1. The gamma function. The hypergeometric functions. Legendre functions , McGraw-Hill (1953)
[4] E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 1. Gewöhnliche Differentialgleichungen , Chelsea, reprint (1971)
How to Cite This Entry:
Confluent hypergeometric equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Confluent_hypergeometric_equation&oldid=46449
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article