Namespaces
Variants
Actions

Confluent hypergeometric function

From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.


Kummer function, Pochhammer function

A solution of the confluent hypergeometric equation

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

The function may be defined using the so-called Kummer series

$$ \tag{2 } \Phi ( \alpha ; \gamma ; z) = \ {} _ {1} F _ {1} ( \alpha , \gamma ; z) = $$

$$ = \ 1 + { \frac \alpha \gamma } { \frac{z}{1!} } + \frac{ \alpha ( \alpha + 1) }{\gamma ( \gamma + 1) } \frac{z ^ {2} }{2! } + \dots , $$

where $ \alpha $ and $ \gamma $ are parameters which assume any real or complex values except for $ \gamma = 0, - 1, - 2 \dots $ and $ z $ is a complex variable. The function $ \Psi ( \alpha ; \gamma ; z ) $ is called the confluent hypergeometric function of the first kind. The second linearly independent solution of equation (1),

$$ \Psi ( \alpha ; \gamma ; z) = \ \frac{\Gamma ( \alpha - \gamma + 1) \Gamma ( \gamma - 1) }{\Gamma ( \alpha ) \Gamma ( 1 - \gamma ) } z ^ {1 - \gamma } \Phi ( \alpha - \gamma + 1 ; 2 - \gamma ; z), $$

$$ \gamma \neq 0 , - 1 , - 2 \dots \ | \mathop{\rm arg} z | < \pi , $$

is called the confluent hypergeometric function of the second kind.

The confluent hypergeometric function $ \Phi ( \alpha ; \gamma ; z ) $ is an entire analytic function in the entire complex $ z $- plane; if $ z $ is fixed, it is an entire function of $ \alpha $ and a meromorphic function of $ \gamma $ with simple poles at the points $ \gamma = 0, - 1 , - 2 ,\dots $. The confluent hypergeometric function $ \Psi ( \alpha ; \gamma ; z ) $ is an analytic function in the complex $ z $- plane with the slit $ ( - \infty , 0 ) $ and an entire function of $ \alpha $ and $ \gamma $.

The confluent hypergeometric function $ \Phi ( \alpha ; \gamma ; z ) $ is connected with the hypergeometric function $ F ( \alpha , \beta , \gamma ; z ) $ by the relation

$$ \Phi ( \alpha ; \gamma ; z) = \ \lim\limits _ {\beta \rightarrow \infty } F \left ( \alpha , \beta , \gamma ; \ { \frac{z} \beta } \right ) . $$

Elementary relationships. The four functions $ \Phi ( \alpha \pm 1 ; \gamma ; z ) $, $ \Phi ( \alpha ; \gamma \pm 1 ; z ) $ are called adjacent (or contiguous) to the function $ \Phi ( \alpha ; \gamma ; z ) $. There is a linear relationship between $ \Phi ( \alpha ; \gamma ; z ) $ and any two functions adjacent to it, e.g.

$$ \gamma \Phi ( \alpha ; \gamma ; z) - \gamma \Phi ( \alpha - 1 ; \gamma ; z) - z \Phi ( \alpha ; \gamma + 1 ; z) = 0. $$

Six formulas of this type may be obtained from the relations between adjacent functions for hypergeometric functions. The successive use of these recurrence formulas yields linear relations connecting the function $ \Phi ( \alpha ; \gamma ; z ) $ with the associated functions $ \Phi ( \alpha + m ; \gamma + n ; z) $, where $ m $ and $ n $ are integers.

Differentiation formulas:

$$ \frac{d ^ {n} }{dz ^ {n} } \Phi ( \alpha ; \gamma ; z) = \ \frac{\alpha \dots ( \alpha + n - 1 ) }{\gamma \dots ( \gamma + n - 1 ) } \Phi ( \alpha + n ; \gamma + n ; z), $$

$$ n = 1 , 2 , . . . . $$

Basic integral representations.

$$ \Phi ( \alpha ; \gamma ; z) = $$

$$ = \ \frac{\Gamma ( \gamma ) }{\Gamma ( \alpha ) \Gamma ( \gamma - \alpha ) } \int\limits _ { 0 } ^ { 1 } e ^ {zt } t ^ {\alpha - 1 } ( 1 - t) ^ {\gamma - \alpha - 1 } dt ,\ \mathop{\rm Re} \gamma > \mathop{\rm Re} \alpha > 0 ; $$

$$ \Psi ( \alpha ; \gamma ; z) = $$

$$ = \ { \frac{1}{\Gamma ( \alpha ) } } \int\limits _ { 0 } ^ \infty e ^ {- zt } t ^ {\alpha - 1 } ( 1 + t) ^ {\gamma - \alpha - 1 } dt ,\ \mathop{\rm Re} \alpha > 0 ,\ \mathop{\rm Re} z > 0 . $$

The asymptotic behaviour of confluent hypergeometric functions as $ z \rightarrow \infty $ can be studied using the integral representations [1], [2], [3]. If $ \gamma \rightarrow \infty $, while $ \alpha $ and $ z $ are bounded, the behaviour of the function $ \Phi ( \alpha ; \gamma ; z) $ is described by formula (2). In particular, for large $ \gamma $ and bounded $ \alpha $ and $ z $:

$$ \Phi ( \alpha ; \gamma ; z) = \ 1 + O ( | \gamma | ^ {-} 1 ) . $$

Representations of functions by confluent hypergeometric functions.

Bessel functions:

$$ J _ \nu ( z) = \ { \frac{1}{\Gamma ( 1 + \nu ) } } \left ( { \frac{z}{2} } \right ) ^ \nu e ^ {- iz } \Phi \left ( \nu + { \frac{1}{2} } ; 2 \nu + 1 ; 2iz \right ) , $$

$$ I _ \nu ( z) = { \frac{1}{\Gamma ( 1 + \nu ) } } \left ( { \frac{z}{2} } \right ) ^ \nu e ^ {- z } \Phi \left ( \nu + { \frac{1}{2} } ; 2 \nu + 1 ; 2z \right ) , $$

$$ K _ \nu ( z) = \sqrt \pi e ^ {- z } ( 2z) ^ \nu \Psi \left ( \nu + { \frac{1}{2} } ; 2 \nu + 1 ; 2z \right ) . $$

Laguerre polynomials:

$$ L _ {n} ^ {( \alpha ) } ( z) = \ \frac{( \alpha + 1) _ {n} }{n! } \Phi (- n ; \alpha + 1 ; z). $$

Probability integrals:

$$ \mathop{\rm erf} ( z) = \ \frac{2z }{\sqrt \pi } \Phi \left ( { \frac{1}{2} } ; { \frac{3}{2} } ; - z ^ {2} \right ) , $$

$$ \mathop{\rm erf} c ( z) = { \frac{1}{\sqrt \pi} } e ^ {- x ^ {2} } \Psi \left ( { \frac{1}{2} } ; { \frac{1}{2} } ; z ^ {2} \right ) . $$

The exponential integral function:

$$ - \mathop{\rm Ei} (- z) = \ e ^ {-} z \Psi ( 1 ; 1 ; z) . $$

The logarithmic integral function:

$$ \mathop{\rm li} ( z) = \ z \Psi ( 1 ; 1 ; - \mathop{\rm ln} z) . $$

Gamma-functions:

$$ \Gamma ( \alpha , z) = \ e ^ {- z } \Psi ( 1 - \alpha ; 1 - \alpha ; z) . $$

Elementary functions:

$$ e ^ {z} = \Phi ( \alpha ; \alpha ; z) , $$

$$ \sin z = e ^ {iz } z \Phi ( 1 ; 2 ; - 2iz) . $$

See also [1], [2], [3], [8].

References

[1] H. Bateman (ed.) A. Erdélyi (ed.) et al. (ed.) , Higher transcendental functions , 2. Bessel functions, parabolic cylinder functions, orthogonal polynomials , McGraw-Hill (1953)
[2] I.S. Gradshtein, I.M. Ryzhik, "Table of integrals, series and products" , Acad. Press (1980) (Translated from Russian)
[3] M. Abramowitz, I.A. Stegun, "Handbook of mathematical functions" , Dover, reprint (1964)
[4] E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 6
[5] A.L. Lebedev, R.M. Fedorova, "Handbook of mathematical tables" , Moscow (1956) (In Russian)
[6] N.M. Burunova, "Handbook of mathematical tables" , Moscow (1959) (In Russian)
[7] A.A. Fletcher, J.C.P. Miller, L. Rosenhead, L.J. Comrie, "An index of mathematical tables" , 1–2 , Oxford Univ. Press (1962)
[8] N.N. Lebedev, "Special functions and their applications" , Prentice-Hall (1965) (Translated from Russian)
How to Cite This Entry:
Confluent hypergeometric function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Confluent_hypergeometric_function&oldid=46450
This article was adapted from an original article by E.A. Chistova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article