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 provethis 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 withinfinite products) we get a decompositionof 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 getthe Taylor expansion of an Eisenstein series, or to evaluate$\zeta (2n)$ for positive integers $n$.