Legendre's Duplication Formula/Proof 2

Theorem

$\forall z \notin \set {-\dfrac n 2: n \in \N}: \map \Gamma z \map \Gamma {z + \dfrac 1 2} = 2^{1 - 2 z} \sqrt \pi \, \map \Gamma {2 z}$

where $\N$ denotes the natural numbers.

Proof

From Gauss Multiplication Formula:

$\ds \prod_{k \mathop = 0}^{n - 1} \map \Gamma {z + \frac k n} = \paren {2 \pi}^{\paren {n - 1} / 2} n^{1/2 - n z} \map \Gamma {n z}$

Substituting $n = 2$ yields:

\(\ds \map \Gamma z \map \Gamma {z + \frac 1 2}\)

\(=\)

\(\ds \paren {2 \pi}^{1 / 2} 2^{1/2 - 2 z} \map \Gamma {2 z}\)

\(\ds \)

\(=\)

\(\ds 2^{1 - 2 z} \sqrt \pi \, \map \Gamma {2 z}\)

$\blacksquare$

Source of Name

This entry was named for Adrien-Marie Legendre.