# examples of infinite products

A classic example is the Riemann zeta function. For $\Re(z)>1$ we have

 $\zeta(z)=\sum_{n=1}^{\infty}\frac{1}{n^{z}}=\prod_{p\text{ prime}}\frac{1}{1-p% ^{-z}}\;.$

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

 $\sin z=z\prod_{n=1}^{\infty}\left(1-\frac{z^{2}}{n^{2}\pi^{2}}\right)$ (1)

where $z$ is an arbitrary complex number. Taking the logarithmic derivative (a frequent move in connection with infinite products) we get a decomposition of the cotangent into partial fractions:

 $\pi\cot\pi z=\frac{1}{z}+\sum_{n=1}^{\infty}\left(\frac{1}{z+n}+\frac{1}{z-n}% \right)\;.$ (2)

The equation (2), in turn, has some interesting uses, e.g. to get the Taylor expansion of an Eisenstein series, or to evaluate $\zeta(2n)$ for positive integers $n$.

Title examples of infinite products ExamplesOfInfiniteProducts 2013-03-22 14:02:32 2013-03-22 14:02:32 mathcam (2727) mathcam (2727) 5 mathcam (2727) Example msc 30E20 ComplexTangentAndCotangent