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The function defined by the formula
 
The function defined by the formula
  
<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/i/i050/i050470/i0504701.png" /></td> </tr></table>
+
$$
 +
I ( x , m )  = \
 +
 
 +
\frac{1}{\Gamma ( m) }
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050470/i0504702.png" /> is the [[Gamma-function|gamma-function]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050470/i0504703.png" /> is an integer, then
+
\int\limits _ { 0 } ^ { x }
 +
e  ^ {-} t t  ^ {m-} 1  dt ,\ \
 +
x \geq  0 ,\  m > 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/i/i050/i050470/i0504704.png" /></td> </tr></table>
+
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:
 
Series representation:
  
<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/i/i050/i050470/i0504705.png" /></td> </tr></table>
+
$$
 +
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:
 
Continued fraction representation:
  
<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/i/i050/i050470/i0504706.png" /></td> </tr></table>
+
$$
 +
I ( x , m ) =
 +
$$
 +
 
 +
$$
 +
= \
 +
1 -
 +
\frac{x  ^ {m} - e  ^ {-} x }{\Gamma ( m
 +
+ 1 ) }
 +
\left \{
 +
\frac{1 \mid  }{\mid  x }
 +
+
 +
\frac{1 - m
 +
\mid  }{\mid  1 }
 +
+
 +
\frac{1 \mid  }{\mid  x }
 +
+
 +
\frac{2 -
 +
m \mid  }{\mid  1 }
 +
+
 +
\frac{2 \mid  }{\mid  x }
 +
+ \dots \right \} .
 +
$$
 +
 
 +
Asymptotic representation for large  $  x $:
  
<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/i/i050/i050470/i0504707.png" /></td> </tr></table>
+
$$
 +
I ( x , m )  = 1 -
  
Asymptotic representation for large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050470/i0504708.png" />:
+
\frac{x  ^ {m-} 1 e  ^ {-} x }{\Gamma ( m) }
  
<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/i/i050/i050470/i0504709.png" /></td> </tr></table>
+
\left \{
 +
\sum _ { i= } 0 ^ { M- }  1
  
Asymptotic representation for large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050470/i05047010.png" />:
+
\frac{( - 1 )  ^ {i} \Gamma ( 1- m+ i ) }{x  ^ {i} \Gamma ( 1- m ) }
  
<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/i/i050/i050470/i05047011.png" /></td> </tr></table>
+
+ O ( x  ^ {-} M )
 +
\right \} .
 +
$$
  
<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/i/i050/i050470/i05047012.png" /></td> </tr></table>
+
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
 
where
  
<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/i/i050/i050470/i05047013.png" /></td> </tr></table>
+
$$
 +
\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]]:
 
Connection with the [[Confluent hypergeometric function|confluent hypergeometric function]]:
  
<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/i/i050/i050470/i05047014.png" /></td> </tr></table>
+
$$
 +
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 }
  
Connection with the [[Laguerre polynomials|Laguerre polynomials]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050470/i05047015.png" />:
+
I ( x , n + \alpha )  = \
 +
( - 1 )  ^ {n} n!
  
<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/i/i050/i050470/i05047016.png" /></td> </tr></table>
+
\frac{\Gamma ( \alpha ) }{\Gamma ( n+ \alpha ) }
 +
 
 +
x ^ {\alpha - 1 } e  ^ {-} x L _ {n} ^ {( \alpha ) } ( x ) .
 +
$$
  
 
Recurrence relation:
 
Recurrence relation:
  
<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/i/i050/i050470/i05047017.png" /></td> </tr></table>
+
$$
 +
m I ( x , m+ 1 ) +
 +
x I ( x , m- 1 )  = \
 +
( x+ m ) I ( x , m ) .
 +
$$
  
 
====References====
 
====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>
 
<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) }
  
====Comments====
+
\int\limits _ { 0 } ^ { x }  t ^ {a - 1 } e  ^ {-} t  d t ,
The following notations are also used:
+
$$
  
<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/i/i050/i050470/i05047018.png" /></td> </tr></table>
+
$$
 +
Q ( a , x )  =
 +
\frac{1}{\Gamma ( a) }
 +
\int\limits _ { x } ^  \infty  t ^ {a - 1 } e  ^ {-} t  d t ,
 +
$$
  
<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/i/i050/i050470/i05047019.png" /></td> </tr></table>
+
with  $  \mathop{\rm Re}  a > 0 $,
 +
$  x \geq  0 $.
 +
The  $  Q $-
 +
function is related to the [[Confluent hypergeometric function|confluent hypergeometric function]]:
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050470/i05047020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050470/i05047021.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050470/i05047022.png" />-function is related to the [[Confluent hypergeometric function|confluent hypergeometric function]]:
+
$$
 +
Q ( a , x )  =
 +
\frac{1}{\Gamma ( a) }
  
<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/i/i050/i050470/i05047023.png" /></td> </tr></table>
+
x  ^ {a} e  ^ {-} x \Psi ( 1 ; a + 1 ; x ) .
 +
$$
  
New asymptotic expansions for both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050470/i05047024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050470/i05047025.png" /> are given in [[#References|[a1]]].
+
New asymptotic expansions for both $  P ( a , x ) $
 +
and $  Q ( a , x ) $
 +
are given in [[#References|[a1]]].
  
 
====References====
 
====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>
 
<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>

Revision as of 22:12, 5 June 2020


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