The Mellin transform
Definition
Let \(\mathbb{R}^{+}\) be the set of positive real numbers. Given a function \(f\) on \(\mathbb{R}^{+}\), define the Mellin transform of \(f\), whenever it makes sense, as follow: \[\mathcal{M}(f)(s)=\int_{0}^{\infty} f(t)t^{s} \frac{dt}{t}. \; (1)\]
The very first example of the Mellin transform I have known is the gamma function, \[\Gamma(s) = \int_{0}^{\infty} e^{-t}t^{s} \frac{dt}{t},\] which is the Mellin transform of the function \(e^{-t}\).