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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, 3 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.

Kartik Sharma - 2 years, 3 months ago

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

Samuel Jones - 2 years, 3 months ago

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It is Lerch Transcendent aka Lerch Phi function

Kartik Sharma - 2 years, 3 months ago

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@Upanshu Gupta

Kartik Sharma - 2 years, 3 months ago

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