# Calculus Challenge 7!

Inspired by Upanshu Gupta!

Generalize:

$\displaystyle \int_{0}^{\infty}{\frac{{x}^{s}{e}^{\alpha x} \ dx}{{(1-\beta {e}^{x})}^{n}}}$

Note by Kartik Sharma
2 years, 10 months ago

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BTW, the integral might have a form like

$$\displaystyle \frac{\Gamma(s+1)}{\Gamma(n-2)} \sum_{k=0}^{n-1}{{(-1)}^{k} {S}_{k} \Phi\left(\beta, s-n+k-2,\alpha\right)}$$ where $${S}_{k}$$ [$${S}_{0} = 1$$] is the $$k$$th symmetric sum of $$\alpha - 1, \alpha - 2, \cdots ,\alpha - n + 1$$

I have not checked this and it might be wrong.

- 2 years, 10 months ago

Can you specify what is $$\Phi (a,b,c)$$ ?

- 2 years, 10 months ago

It is Lerch Transcendent aka Lerch Phi function

- 2 years, 10 months ago