I know that Euler proved that
$\displaystyle z! = \int_{0}^{1} \left(- \ln\left(t\right)\right)^z dt$
How can this be proven?
2026-03-26 22:13:19.1774563199
Euler Representation of the factorial
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With definition of Gamma function $$\Gamma(x)=\int_0^\infty e^{-t}t^{x-1}\ dt$$ let $e^{-t}=u$ then $$\Gamma(x)=\int_1^0 u(-\ln u)^{x-1}\dfrac{-1}{u} du$$ and so $$x!=\Gamma(x+1)=\int_0^1 (-\ln u)^{x} du$$