The Mellin transform is an integral transform defined as follows: $$ F(s) = \int_0^\infty f(t) t^{s-1} \, dt $$ Intuitively, it may be viewed as a continuous analogue of a power series -- instead of synthetizing a function by summing multiples of integer powers, we integrate over all real powers. This transform is closely related to the Laplace transform -- if we make a change of variables $t = e^{-r}$ and define $g$ by $f(e^{-r}) = g(r)$ , then the above integral becomes $$ F(s) = -\int_{-\infty}^{+\infty} g(r) e^{-rs} \, dr , $$ which is a bilateral Laplace transform.

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