Legendre's Duplication Formula
Theorem
Let $\Gamma$ denote the gamma function.
Then:
- $\displaystyle \forall z \notin -\frac 1 2 \N_0: \Gamma \left({z}\right) \Gamma \left (z + \frac 1 2 \right) = 2^{1-2z} \sqrt \pi \Gamma \left({2 z}\right)$
where $N_0 = \N \cup \left\{{0}\right\}$.
What's $-\frac 1 2 \N_0$?
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Proof
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Source of Name
This entry was named for Adrien-Marie Legendre.
