Incomplete gamma function

The upper incomplete gamma function is $\gamma (x,y)={\int }_y^{\infty }{t}^{x-1}{e}^{-t}dt$ under the convention used here.

For $a>0$, this integral is an entire function of $x$: the exponential series converges uniformly on bounded subsets after multiplication by the integrable bound $e^{|x|}{u}^{a-1}$. ThusFor positive real $x$, a change of variables gives $F_a(x)=x^{-a}\gamma (a,x)$, where here $\gamma (a,x)={\int }_0^x{e}^{-t}{t}^{a-1}dt$ denotes the lower incomplete gamma function. The singular-looking factor $x^{-a}$ is removed by the integral over $[0,1]$. Applying integration by parts to the derivative of $u^a{e}^{-xu}$ gives $x{F}_a^{\prime }+a{F}_a=e^{-x}$; differentiating and eliminating $e^{-x}$ proves $x{F}_a^{\prime \prime }+(a+1+x)F_a^{\prime }+a{F}_a=0$.

Repeated integration by parts generates successive inverse powers of a large endpoint, with a remainder expressed by the same integral with shifted parameters.

For fixed $x$ and $y\to \infty$,

Forrepeated integration by parts gives

For fixed finite $y$ and $x\to \infty$, the omitted integral from zero to $y$ is negligible, so $\gamma (x,y)\sim \Gamma (x)$ and Stirling's formula applies.

When $y=\lambda x$ with $\lambda >1$, the integral is dominated by its lower endpoint. Setting $t=y+s$ and applying Watson's lemma gives coefficients rational in $\lambda -1$.