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→Periodic zeta function: Incorrect eigenvalue on F(s;kq) |
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for integer ''k'' ≥ 1, and is sometimes called '''Gauss's multiplication formula''', in honour of [[Carl Friedrich Gauss]]. The multiplication theorem for the gamma functions can be understood to be a special case, for the trivial [[Dirichlet character]], of the [[Chowla–Selberg formula]].
==Sine function ==
Formally similar duplication formulas hold for the sine function, which are rather simple consequences of the [[List of trigonometric identities|trigonometric identities]]. Here one has the duplication formula
:<math>
\sin(\pi x)\sin\left(\pi\left(x+\frac{1}{2}\right)\right) = \frac{1}{2}\sin(2\pi x)
</math>
and, more generally, for any integer ''k'', one has
:<math>
\sin(\pi x)\sin\left(\pi\left(x+\frac{1}{k}\right)\right) \cdots \sin\left(\pi\left(x+\frac{k-1}{k}\right)\right) = 2^{1-k} \sin(k \pi x)
</math>
==Polygamma function, harmonic numbers==
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