Incomplete gamma-function
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 |
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