Gamma(z) as integral of a power of log(1/t) for Re(z) greater than 0

Theorem

The following formula holds: $$\Gamma(z) = \displaystyle\int_0^1 \log \left( \dfrac{1}{t} \right)^{z-1} \mathrm{d}t,$$ where $\Gamma$ denotes the gamma function and $\log$ denotes the logarithm.

References

• 1953: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi: Higher Transcendental Functions Volume I ... (previous) ... (next): §1.1 (1)