I'm wondering how this formula was derived. I'm looking for a proof. My guess is that there was a substitution made, and through contour integration or something, it was shown that the limits don't change even when $$\phi(n)$$ isn't real. Here's the formula
$$\begin{array}{l} \int\limits ^{\infty }_{0} t^{s-1} \cdot f( t) dt=\Gamma ( s) \cdot \phi ( -s) ,\text{ if } f( t) =\sum\limits ^{\infty }_{n=0}\frac{\phi ( n)}{n!}( -t)^{n}\\ \end{array}$$
And I think $$\phi(n)$$ can be any sequence.