The incomplete gamma function (upper) is defined as $\Gamma(a,x)=\int_x^{\infty}t^{a-1}e^{-t}dt$. Is there a series expansion for small non-integer $a << 1$ and small $x$?
Series Expansion of the Incomplete Gamma Function
2
$\begingroup$
sequences-and-series
gamma-function
-
1You want an expansion around $x=0$ assuming $a \in (0,1)$ ? Did you notice that $\frac{\partial}{\partial x} \Gamma(a,x) = -x^{a-1} e^{-x}$ ? – 2017-02-27
1 Answers
2
Thanks to this question, we know that
$$\Gamma(a,x)=\Gamma(a)-\sum_{n=0}^\infty\frac{(-1)^na^{s+n}}{n!(x+n)}$$
Which follows from the Taylor expansion of $e^x$.
-
0Of course, apply any expansion of the good 'ole Gamma function you'd like. – 2017-02-28