# examples of infinite products

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

$$\zeta (z)=\sum _{n=1}^{\mathrm{\infty}}\frac{1}{{n}^{z}}=\prod _{p\text{prime}}\frac{1}{1-{p}^{-z}}.$$ |

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

$$\mathrm{sin}z=z\prod _{n=1}^{\mathrm{\infty}}\left(1-\frac{{z}^{2}}{{n}^{2}{\pi}^{2}}\right)$$ | (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^{}:

$$\pi \mathrm{cot}\pi z=\frac{1}{z}+\sum _{n=1}^{\mathrm{\infty}}\left(\frac{1}{z+n}+\frac{1}{z-n}\right).$$ | (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 |
---|---|

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 |