Tan(pi*x) identity

Dec 2016
62
10
-
Is there any identity for $\tan(\pi z)$ in terms of closed analytical formulas removing the factor $\pi$ from the argument?

I only know that
\begin{eqnarray}
\sin(\pi z)=\frac{\pi}{\Gamma(z)\Gamma(1-z)}
\end{eqnarray}

so could work as well with a formula for $\cos(\pi z)$ in the same way?
 

romsek

Math Team
Sep 2015
2,751
1,531
USA
$\cos(\pi z) = \sin\left(\pi \left(z + \dfrac 1 2 \right)\right)$

$\cos(\pi z ) = \dfrac{\pi}{\Gamma\left(z+\frac 1 2 \right)\Gamma\left(\frac 1 2 - z\right)}$

$\tan(\pi z) = \dfrac{\Gamma\left(z+\frac 1 2 \right)\Gamma\left(\frac 1 2 - z\right)}{\Gamma(z)\Gamma(1-z)}$
 
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