I am trying to evaluate $$\int_{x=0}^{\infty}\left(\frac{1}{\sqrt{x}}(1-e^{-x})\right)^{M-1}e^{-x}(1+sx)^{-N}dx,$$ where $s>0$, $M$ and $N$ are positive integers.
But seem that the above integral does not not have a closed form expression.
Then I step back and want only an expression for very large $s$, i.e. as $s \rightarrow \infty $, but seems that dominate Dominated convergence theorem theorem also does not work for it.
Can anyone tell me how can I deal with this integral?
Thanks.
Not for a general M, no. Not unless you're willing to throw hypergeometric functions into the mix. But it does for each particular value of M.
Yes. The whole idea is to write $I(a)=\displaystyle\int_0^\infty\bigg(\frac{1-e^{-x}}{\sqrt x}\bigg)^m\frac{e^{-x}}{a+sx}~dx,~$ and then to notice that
your integral is nothing else than a multiple of $I^{(n)}(1)$, where $m=M-1$, and $n=N-1$. For
$m=0$ we have $~I(a)=\dfrac{\exp\bigg(\dfrac as\bigg)~\Gamma\bigg(0,~\dfrac as\bigg)}s~,$ and for $m=1$ we get $~I(a)=\pi~\dfrac{\exp\bigg(\dfrac as\bigg)}{\sqrt{as}}\cdot$
$\cdot\bigg[\text{erfc}\bigg(\sqrt{\dfrac as}\bigg)-\exp\bigg(\dfrac as\bigg)~\text{erfc}\bigg(\sqrt{2~\dfrac as}\bigg)\bigg],~$ etc.