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For instance, take the gamma function. Somehow there is a way to use the identity $f(z+1)= z f(z)$ and the condition $f(1)=1$ to derive the Euler/Laplace integral representation of the gamma function $\Gamma(z)$ using complex analysis as a solution to the functional equation, but how is that done? From what I have seen it is some kind of complicated analytical process involving contour integrals, can you apply a similar process to other functional identities?

Simply Beautiful Art
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TeXnichal
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    Even smoothness, concavity, and the functional equation are not sufficient to define the Gamma function, as pointed out in the above duplicate, supposing your question is specifically about the Gamma function. If it is not, then your question is probably unclear and still answerable by the above link (as no, in general you cannot define a function uniquely from just a functional equation on $\mathbb C$ without more restrictions). – Simply Beautiful Art Sep 21 '19 at 01:17
  • I don't see how this is a duplicate of a factorial, this is clearly about conventional techniques of analyzing a functional equation in order to derive a result, using the gamma function as an example since it is a classic. Whoever said this question was the same was mistaken. I also don't see hardly the functional identity nor a single contour integral. You are given a functional equation and maybe a few values and characteristics, and given those, how does one use complex analysis to find the solution? I don't see how that post addresses that, even for the gamma function. – TeXnichal Sep 21 '19 at 06:36
  • It addresses the fact that you most likely need a lot more conditions. Even if it weren't a duplicate, you didn't make this clear enough in the question and it is still too broad to answer anyways. – Simply Beautiful Art Sep 21 '19 at 13:22
  • If you need more conditions then one can simply point that out in the answer, it wouldn't be that difficult to do for an accredited professional who has any merit in trying to answer this question. – TeXnichal Sep 21 '19 at 19:42
  • No, because that's not an answer, but a comment as to why the question can't be answered. – Simply Beautiful Art Sep 22 '19 at 12:31

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