I don't see how that would change the limit of integration. In order to make the substitution, I did the following:
t^{z-1} = n^{z-1} (\frac{t}{t})^{z-1} = n^{z-1} s^{z-1}
(1-\frac{t}{n})^n = (1-s)^n
ds=\frac{1}{n}dt \rightarrow dt=nds
All that gives me \Gamma _n(z)=n^z\int_0^n...
I'm working through rewriting the gamma function as an infinite product, but my question is just about a specific substitution that was made in my textbook. They took the equation:
\Gamma_ n(z)=\int_0^n t^{z-1} (1-\frac{t}{n}) ^ndt for Re(z)>0 and n greater than or equal to 1.
and made the...