View source for 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|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|confluent hypergeometric function]]:

$$ 
I ( x , m )  = \ 

\frac{x  ^ {m} }{\Gamma ( m+ 1 ) }

{} _ {1} F _ {1} ( m , m+ 1 ;  - x ) .
$$

Connection with the [[Laguerre polynomials|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Abramowitz,   I.A. Stegun,   "Handbook of mathematical functions" , Dover, reprint  (1973)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.I. Pagurova,   "Tables of the incomplete gamma-function" , Moscow  (1963)  (In Russian)</TD></TR></table>

====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|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 [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.M. Temme,   "The asymptotic expansion of the incomplete gamma functions"  ''SIAM J. Math. Anal.'' , '''10'''  (1979)  pp. 757–766</TD></TR></table>