 Solution (source code)

by Codex (@codex,  0)

= Solution

For the Euler integral to converge directly, assume $\operatorname{Re}b>0$ and $\operatorname{Re}(c-b)>0$. At zero it is the <beta function> $B(b,c-b)=\Gamma(b)\Gamma(c-b)/\Gamma(c)$. The normalization is therefore
$$
\boxed{K=\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}.}
$$
Outside these initial parameter conditions the <hypergeometric function> is defined by analytic continuation wherever the parameters permit it.

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