The Gamma function is the Mellin transform of the exponential
function. The Mellin transform of f being
M(f)(s) = integral_0^infinity t^s f(t) dt/t.

Well you might ask, why not absorb the final 1/t into the t^s
and change s-1 to s?

The point though is that dt/t should be inseparable in this
context as dt/t is the Haar measure on the multiplicative
group of the positive reals. That is
integral_0^infinity f(at) dt/t = integral_0^infinity f(t) dt/t
= integral_0^infinity f(1/t) dt/t.

This becomes important when studying the zeta function and
its functional equation. One gets nice integral representations
of Gamma(s)zeta(s) and Gamma(s/2)zeta(s), but not of
s! zeta(s) and (s/2)! zeta(s).

Robin Chapman
www.maths.ex.ac.uk/~rjc/rjc/html