Calculus Challenge 7!

Inspired by Upanshu Gupta!

Generalize:

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

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

@Upanshu Gupta

Kartik Sharma - 5 years, 9 months ago

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 - 5 years, 9 months ago

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

Samuel Jones - 5 years, 9 months ago

It is Lerch Transcendent aka Lerch Phi function

Kartik Sharma - 5 years, 9 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$