# Incomplete gamma-function

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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:

$$I ( x , m ) =$$

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

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=47326
This article was adapted from an original article by V.I. Pagurova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article