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Any chance i can find a definite integral for this complex function?

\begin{equation} f_n(z)=\frac{2^{-\frac{1}{2} ( \sin (\pi z) ) ( \sin (2 \pi n))+2 n-2} \pi ^{\frac{1}{2} ( \sin (\pi z) ) ( \sin (2 \pi n) )} \Gamma \left(\frac{z-1}{2}+n\right)}{\Gamma \left(\frac{1}{2} (z-2 n+3)\right)} \quad n\in\mathbb{Z} \end{equation}

More specifically, this is the analytical continuation of $\frac{(z+2 n-3)\text{!!}}{(z-2 n+1)\text{!!}}$ to $z\in\mathbb{C}$ which i need to integrate and then find the value of it at $z\in\mathbb{Z}$.

Any sugestions are welcome :D

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  • $\begingroup$ For a given value of $n>0$, the integrand is just a polynomial. But which one ? $\endgroup$ Jul 4 at 3:19
  • $\begingroup$ I am quite curious about the background of this problem. $\endgroup$
    – TravorLZH
    Jul 4 at 6:02

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