Who derived $\int_0^\infty e^{-nx}x^{s-1}dx = \Pi(s-1)/n^s$?

Question

Does anyone know the name of the paper in which this equation first appeared? Thank you! $$\int_0^\infty e^{-nx}x^{s-1}dx = \Pi(s-1)/n^s$$

Answer 1

It was (almost certainly) Euler. For example, we find it in E675, De valoribus integralium a termino variabilis $x=0$ usque ad $x=\infty$ extensorum (1781). A translation of this paper may be found on arXiv. At the bottom of p. 3 of the translation one finds

§8. I first put $x=ky$, and because both terms of the integral stay the same, it will be $$k^n \int y^{n−1}\partial y \cdot e^{−ky}= \Delta, $$ where this formula is also extended from $y= 0$ up to $y=\infty$; then dividing this by $k^n$ we will have $$ \int y^{n−1}\partial y \cdot e^{−ky}=\frac{\Delta}{k^n} $$,where it should however be noted that no negative numbers can be taken for $k$, as otherwise the formula $e^{−ky}$ would no longer vanish in the case $y=\infty$;and these are the only values which ought to be excluded here, so that even imaginary values could be used in place of $k$, and then I pursued these laborious integrations.

(And yes, Euler does consider non-integer $n$ in the paper: he discusses $n=1/2$ later on.)

Answer 2

That formula is easily seen to be equivalent to the integral definition of the Gamma function, introduced by Euler in a letter to Goldbach in 1730. (Note that $\Pi(s-1)=\Gamma(s)$.)

Source: Wikipedia.

Answer 3

I suppose that it was in Riemann's On the number of primes less than a given magnitude. It's the second formula that appears there.