Euler's Reflection Formula

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Theorem

Let $\Gamma$ denote the gamma function.

Then:

$\forall z \notin \Z: \map \Gamma z \, \map \Gamma {1 - z} = \dfrac \pi {\map \sin {\pi z} }$


Corollary

$\forall z \notin \Z: \paren {-z}! \, \map \Gamma z = \dfrac \pi {\map \sin {\pi z} }$


Proof

We have the Weierstrass products:

$\displaystyle \map \sin {\pi z} = \pi z \prod_{n \mathop \ne 0} \paren {1 - \frac z n} \map \exp {\frac z n}$


From the Weierstrass form of the Gamma function:

$\displaystyle \frac 1 {\map \Gamma z} = z e^{\gamma z} \prod_{n \mathop = 1}^\infty \paren {1 + \frac z n} \map \exp {-\frac z n}$

from which:

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

whence:

\(\displaystyle \map \Gamma z \, \map \Gamma {1 - z}\) \(=\) \(\displaystyle -z \, \map \Gamma z \, \map \Gamma {-z}\) Gamma Difference Equation
\(\displaystyle \) \(=\) \(\displaystyle \frac \pi {\map \sin {\pi z} }\)

$\blacksquare$


Source of Name

This entry was named for Leonhard Paul Euler.


Sources