examples of infinite products

A classic example is the Riemann zeta functionDlmfDlmfMathworldPlanetmath. For (z)>1 we have

ζ(z)=n=11nz=p prime11-p-z.

With the help of a Fourier seriesMathworldPlanetmath, or in other ways, one can prove this infinite product expansion of the sine function:

sinz=zn=1(1-z2n2π2) (1)

where z is an arbitrary complex numberPlanetmathPlanetmath. Taking the logarithmic derivativeMathworldPlanetmath (a frequent move in connection with infinite products) we get a decomposition of the cotangentMathworldPlanetmathPlanetmath into partial fractionsPlanetmathPlanetmath:

πcotπz=1z+n=1(1z+n+1z-n). (2)

The equation (2), in turn, has some interesting uses, e.g. to get the Taylor expansionMathworldPlanetmath of an Eisenstein seriesMathworldPlanetmath, or to evaluate ζ(2n) for positive integers n.

Title examples of infinite products
Canonical name ExamplesOfInfiniteProducts
Date of creation 2013-03-22 14:02:32
Last modified on 2013-03-22 14:02:32
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 5
Author mathcam (2727)
Entry type Example
Classification msc 30E20
Related topic ComplexTangentAndCotangent