Namespaces
Variants
Actions

Incomplete gamma-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.


The function defined by the formula

$$ I ( x , m ) = \ \frac{1}{\Gamma ( m) } \int\limits _ { 0 } ^ { x } e ^ {-t} t ^ {m-1} dt ,\ \ x \geq 0 ,\ m > 0 , $$

where $ \Gamma ( m) = \int _ {0} ^ \infty e ^ {-t} t ^ {m-1} dt $ is the gamma-function. If $ n \geq 0 $ is an integer, then

$$ I ( x , n+ 1 ) = \ 1 - e ^ {-x} \sum _ { m= 0}^ { n } \frac{x ^ {m} }{m ! } . $$

Series representation:

$$ I ( x , m ) = \ \frac{e ^ {-x} x ^ {m} }{\Gamma ( m+ 1 ) } \left \{ 1+ \sum _ { k= 1}^\infty \frac{x ^ {k} }{( m+ 1 ) \dots ( m+ k ) } \right \} . $$

Continued fraction representation:

$$ I ( x , m ) = $$

$$ = \ 1 - \frac{x ^ {m-1} e ^ {-x} }{\Gamma ( m + 1 ) } \left \{ \frac{1 \mid }{\mid x } + \frac{1 - m \mid }{\mid 1 } + \frac{1 \mid }{\mid x } + \frac{2 - m \mid }{\mid 1 } + \frac{2 \mid }{\mid x } + \dots \right \} . $$

Asymptotic representation for large $ x $:

$$ I ( x , m ) = 1 - \frac{x ^ {m-1} e ^ {-x} }{\Gamma ( m) } \left \{ \sum _ { i= 0}^{ M- 1} \frac{( - 1 ) ^ {i} \Gamma ( 1- m+ i ) }{x ^ {i} \Gamma ( 1- m ) } + O ( x^{-M} ) \right \} . $$

Asymptotic representation for large $ m $:

$$ I ( x , m ) = \ \Phi ( 2 \sqrt x - \sqrt m- 1 ) + O ( m ^ {-1/2} ) , $$

$$ I ( x , m ) = \Phi \left [ 3 \sqrt m \left ( \left ( \frac{x}{m} \right ) ^ {1/3} - 1 + \frac{1}{9m} \right ) \right ] + O ( m ^ {-1} ) , $$

where

$$ \Phi ( z) = \ \frac{1}{\sqrt {2 \pi } } \int\limits _ {- \infty } ^ { z } e ^ {- t ^ {2} / 2 } dt . $$

Connection with the confluent hypergeometric function:

$$ I ( x , m ) = \ \frac{x ^ {m} }{\Gamma ( m+ 1 ) } {} _ {1} F _ {1} ( m , m+ 1 ; - x ) . $$

Connection with the Laguerre polynomials $ L _ {n} ^ {( \alpha ) } ( x) $:

$$ \frac{\partial ^ {n+1} }{\partial x ^ {n+1} } I ( x , n + \alpha ) = \ ( - 1 ) ^ {n} n! \frac{\Gamma ( \alpha ) }{\Gamma ( n+ \alpha ) } x ^ {\alpha - 1 } e ^ {-x} L _ {n} ^ {( \alpha ) } ( x ) . $$

Recurrence relation:

$$ m I ( x , m+ 1 ) + x I ( x , m- 1 ) = \ ( x+ m ) I ( x , m ) . $$

References

[1] M. Abramowitz, I.A. Stegun, "Handbook of mathematical functions" , Dover, reprint (1973)
[2] V.I. Pagurova, "Tables of the incomplete gamma-function" , Moscow (1963) (In Russian)

Comments

The following notations are also used:

$$ P ( a , x ) = \frac{1}{\Gamma ( a) } \int\limits _ { 0 } ^ { x } t ^ {a - 1 } e ^ {-t} d t , $$

$$ Q ( a , x ) = \frac{1}{\Gamma ( a) } \int\limits _ { x } ^ \infty t ^ {a - 1 } e ^ {-t} d t , $$

with $ \mathop{\rm Re} a > 0 $, $ x \geq 0 $. The $ Q $- function is related to the confluent hypergeometric function:

$$ Q ( a , x ) = \frac{1}{\Gamma ( a) } x ^ {a} e ^ {-x} \Psi ( 1 ; a + 1 ; x ) . $$

New asymptotic expansions for both $ P ( a , x ) $ and $ Q ( a , x ) $ are given in [a1].

References

[a1] N.M. Temme, "The asymptotic expansion of the incomplete gamma functions" SIAM J. Math. Anal. , 10 (1979) pp. 757–766
How to Cite This Entry:
Incomplete gamma-function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Incomplete_gamma-function&oldid=51819
This article was adapted from an original article by V.I. Pagurova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article