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[parent] examples of infinite products (Example)

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

$\displaystyle \zeta(z)=\sum_{n=1}^\infty\frac{1}{n^z}= \prod_{p\text{ prime}}\frac{1}{1-p^{-z}}\;.$
With the help of a Fourier series, or in other ways, one can prove this infinite product expansion of the sine function:
$\displaystyle \sin z=z\prod_{n=1}^\infty\left(1-\frac{z^2}{n^2\pi^2}\right)$ (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:
$\displaystyle \pi\cot\pi z=\frac{1}{z}+\sum_{n=1}^\infty \left(\frac{1}{z+n}+\frac{1}{z-n}\right)\;.$ (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$.



"examples of infinite products" is owned by mathcam. [ full author list (2) | owner history (1) ]
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See Also: complex tangent and cotangent


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Cross-references: integers, positive, Eisenstein series, Taylor expansion, equation, partial fractions, cotangent, logarithmic derivative, complex number, function, sine, product, infinite, Fourier series, Riemann zeta function
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This is version 2 of examples of infinite products, born on 2003-10-16, modified 2007-01-11.
Object id is 5391, canonical name is ExamplesOfInfiniteProducts.
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Classification:
AMS MSC: 30E20 (Functions of a complex variable :: Miscellaneous topics of analysis in the complex domain :: Integration, integrals of Cauchy type, integral representations of analytic functions)
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