Euler's Reflection Formula

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Theorem

Let $\Gamma$ denote the gamma function.

Then:

$\forall z \notin \Z: \Gamma \left({z}\right) \Gamma \left({1 - z}\right) = \dfrac \pi {\sin \left({\pi z}\right)}$


Corollary

$\forall z \notin \Z: \left({-z}\right)! \, \Gamma \left({z}\right) = \dfrac \pi {\sin \left({\pi z}\right)}$


Proof

We have the Weierstrass products:

$\displaystyle \sin \left({\pi z}\right) = \pi z \prod_{n \mathop \ne 0} \left({1 - \frac z n}\right) \exp \left({\frac z n}\right)$


From the Weierstrass form of the Gamma function:

$\displaystyle \frac 1 {\Gamma \left({z}\right)} = z e^{\gamma z} \prod_{n \mathop = 1}^\infty \left({1 + \frac z n}\right) \exp \left({-\frac z n}\right)$

from which:

\(\displaystyle \dfrac 1 {-z \Gamma \left({z}\right) \Gamma \left({-z}\right)}\) \(=\) \(\displaystyle \frac {-z^2 \exp \left({\gamma z}\right) \exp \left({-\gamma z}\right)} {-z} \prod_{n \mathop = 1}^\infty \left({1 + \frac z n}\right) \left({1 - \frac z n}\right) \exp \left({\frac z n}\right) \exp \left({-\frac z n}\right)\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle z \prod_{n \mathop = 1}^\infty \left({1 - \frac {z^2} {n^2} }\right)\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {\sin \left({\pi z}\right)} \pi\) $\quad$ Euler Formula for Sine Function $\quad$

whence:

\(\displaystyle \Gamma \left({z}\right) \Gamma \left({1 - z}\right)\) \(=\) \(\displaystyle -z \Gamma \left({z}\right) \Gamma \left({-z}\right)\) $\quad$ Gamma Difference Equation $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac \pi {\sin \left({\pi z}\right)}\) $\quad$ $\quad$

$\blacksquare$


Source of Name

This entry was named for Leonhard Paul Euler.


Sources