Multiplication theorem: Difference between revisions

Revision as of 06:06, 2 December 2014

Revision as of 11:06, 16 June 2015

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</math>

for integer ''k'' &ge; 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 character]], of the [[Chowla–Selberg formula]].

This implies a variation of [[Stirling's approximation]] that is exact --- a '''Stirling identity''' ---

:<math>

\exp\left(\int_z^{z+1}\log\Gamma(x)\,dx\right) = \sqrt{2\pi} z^ze^{-z}.

</math>

The left side is the '''geometric mean''' of <math>\log\Gamma</math> over the line segment <math>[z,z+1]</math>.

==Polygamma function==