## You are here

Homeexamples of infinite products

## Primary tabs

# 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 $n$.

Related:

ComplexTangentAndCotangent

Major Section:

Reference

Type of Math Object:

Example

## Mathematics Subject Classification

30E20*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections