Yeah. It’s artificially introduced (why the s-1 power?) and is basically just confusing. Gamma function isn’t really something I’ve had reason to use myself, so I’m just going on the fact that I’ve heard lots of people complain about this and never anyone defending it, to conclude that it really is as dumb as it looks.
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)).
Yeah. It’s artificially introduced (why the s-1 power?) and is basically just confusing. Gamma function isn’t really something I’ve had reason to use myself, so I’m just going on the fact that I’ve heard lots of people complain about this and never anyone defending it, to conclude that it really is as dumb as it looks.
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)).