The first one follows from a change of variable of the integral representation of the Gamma function (Abel, 1823), giving a Mellin transform which can be expressed in different ways as a double integral (Sondow, 2005).
The function has an integral representation in terms of the Mellin transform as
Fourier transform | Laplace transform | Fast Fourier transform | Extract, transform, load | Robert Mellin | Hilbert transform | Fourier transform infrared spectroscopy | fast Fourier transform | Discrete cosine transform | transform | Mellin transform | discrete cosine transform | Bilinear transform | Z-transform | Y-Δ transform | Wavelet transform modulus maxima method | Mellin de Saint-Gelais | Hough transform | Gabor–Wigner transform | Gabor transform | fractional Fourier transform | Fourier transform spectroscopy | Fourier Transform | fast fourier transform | Discrete Hartley transform | Adaptive Transform Acoustic Coding (ATRAC) | Adaptive Transform Acoustic Coding |