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.

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    $\begingroup$ Do you have a source for this formula? $\endgroup$ – Lukas Kofler Aug 12 at 22:39
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    $\begingroup$ It is Ramanujan's master theorem. You need stronger conditions (decay, analyticity), as it is it is not true. See math.stackexchange.com/questions/3281841/… $\endgroup$ – reuns Aug 12 at 23:13

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