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Euler's Reflection Formula Derivation 📂Functions

Euler's Reflection Formula Derivation

Formulas

For non-integer $p$, $$ {\Gamma (1-p) \Gamma ( p )} = { {\pi} \over {\sin \pi p } } $$

Description

It is the most famous formula among the formulas using the Gamma function.

A useful result that can be obtained from the reflection formula is $ \Gamma ( { 1 \over 2} ) = \sqrt{\pi}$. Perhaps that’s why? The name “reflection formula” is said to have been derived from reflecting on $\frac{1}{2}$.

Derivation

Weierstrass’s infinite product: $$ {1 \over \Gamma (p)} = p e^{\gamma p } \prod_{n=1}^{\infty} \left( 1 + {p \over n} \right) e^{- {p \over n} } $$

$$ \begin{align*} {{1} \over {\Gamma (p)}} \cdot { 1 \over { \Gamma ( -p )}} =& p e^{\gamma p } \prod_{n=1}^{\infty} \left( 1 + {p \over n} \right) e^{- {p \over n} } \cdot (-p) e^{- \gamma p } \prod_{n=1}^{\infty} \left( 1 - {p \over n} \right) e^{ {p \over n} } \\ =& -p^2 \prod_{n=1}^{\infty} \left( 1 - {p^2 \over n^2} \right) \end{align*} $$ Meanwhile, since ${ \Gamma ( 1-p )} = -p \Gamma (-p)$, $$ { 1 \over {\Gamma (1-p) \Gamma ( p )} } = p \prod_{n=1}^{\infty} \left( 1 - {p^2 \over n^2} \right) $$

Euler’s representation of the sinc function: $$ {{\sin \pi x} \over {\pi x}} = \prod_{n=1}^{\infty} \left( 1 - {{x^2} \over { n^2}} \right) $$

By fine-tuning the Euler representation of the sinc function, the desired formula is obtained.