Euler's Reflection Formula/Corollary
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Corollary to Euler's Reflection Formula
Let $\Gamma$ denote the gamma function.
Then:
- $\forall z \notin \Z: \paren {-z}! \, \map \Gamma z = \dfrac \pi {\map \sin {\pi z} }$
Proof
\(\ds \dfrac \pi {\map \sin {\pi z} }\) | \(=\) | \(\ds \map \Gamma z \map \Gamma {1 - z}\) | Euler's Reflection Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \Gamma {-z + 1} \map \Gamma z\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-z}! \, \map \Gamma z\) | Gamma Function Extends Factorial |
$\blacksquare$
Source of Name
This entry was named for Leonhard Paul Euler.
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.5$: Permutations and Factorials