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:

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

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