Derivation of the Legendre Duplication Formula
Formula
$$ \Gamma (2r) = {{2^{ 2r - 1} } \over { \sqrt{ \pi } } } \Gamma \left( r \right) \Gamma \left( {{1} \over {2}} + r \right) $$
Description
While the splitting shape may not be pretty, the fact that factors can be divided into smaller ones is certainly useful. The derivation itself is not very difficult if one uses an auxiliary lemma derived from the beta function.
Derivation
For $$ B(p,q) = {{\Gamma (p) \Gamma (q)} \over {\Gamma (p+q) }} = \int_{0}^{1} t^{p-1} (1-t)^{q-1} dt $$, if it is said that $r:= p=q$ $$ {{\Gamma (r) \Gamma (r)} \over {\Gamma (2r) }} = \int_{0}^{1} t^{r-1} (1-t)^{r-1} dt $$ When substituting $\displaystyle t = {{1+s} \over {2}}$ for $\lambda (s) := \left( 1 - s^2 \right)^{r-1}$, since it is an even function, $$ \begin{align*} {{\Gamma (r) \Gamma (r)} \over {\Gamma (2r) }} =& {{1} \over {2}} \int_{-1}^{1} \left( {{1+s} \over {2}} \right)^{r-1} \left( {{1-s} \over {2}} \right)^{r-1} ds \\ =& {{1} \over {2^{1 + 2(r-1)} }} \int_{-1}^{1} \left( 1 - s^2 \right)^{r-1} ds \\ =& 2^{1 - 2r} \cdot 2 \int_{0}^{1} \left( 1 - s^2 \right)^{r-1} ds \end{align*} $$
Corollary of the trigonometric expression of the beta function: $$ B(x,y) = 2 \int_{0}^{1} t^{2x-1} \left( 1 - t^2 \right)^{y-1} dt $$
By substituting $\displaystyle x = {{1} \over {2}}$ and $y = r$ into the above formula, $$ B \left( {{1} \over {2}} , r \right) = 2 \int_{0}^{1} \left( 1 - t^2 \right)^{r-1} dt $$ Therefore, $$ {{\Gamma (r) \Gamma (r)} \over {\Gamma (2r) }} = 2^{1 - 2r} B \left( {{1} \over {2}} , r \right) = 2^{1 - 2r} {{\Gamma \left( {{1} \over {2}} \right) \Gamma (r)} \over {\Gamma \left( {{1} \over {2}} + r \right) }} $$ is obtained. Since from the reflection formula $\displaystyle \Gamma \left( {1 \over 2} \right) = \sqrt{\pi}$ $$ {{\Gamma (r)} \over {\Gamma (2r) }} = 2^{1 - 2r} {{\sqrt{\pi} } \over {\Gamma \left( {{1} \over {2}} + r \right) }} $$ Organizing with respect to $\Gamma (2r)$ yields $$ \Gamma (2r) = {{2^{2r-1} } \over { \sqrt{ \pi } } } \Gamma \left( r \right) \Gamma \left( {{1} \over {2}} + r \right) $$
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