The t^(s-1) in the gamma function should be thought of as the product of t^s dt/t. This is a standard part of the Mellin transform. The dt/t is invariant under multiplication, which is a sensible thing to ask for since the domain of integration (0,infinity) is preserved by scaling, but not by the translations that preserve dt.
In other words, dt/t = d(log t) and it’s telling you to change variables: the gamma function is the Laplace (or Fourier) transform of exp(-exp(u)).
The t^(s-1) in the gamma function should be thought of as the product of t^s dt/t. This is a standard part of the Mellin transform. The dt/t is invariant under multiplication, which is a sensible thing to ask for since the domain of integration (0,infinity) is preserved by scaling, but not by the translations that preserve dt.
In other words, dt/t = d(log t) and it’s telling you to change variables: the gamma function is the Laplace (or Fourier) transform of exp(-exp(u)).