examples of infinite products
A classic example is the Riemann zeta function.
For we have
With the help of a Fourier series, or in other ways, one can prove
this infinite product expansion of the sine function:
(1) |
where 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
:
(2) |
The equation (2), in turn, has some interesting uses, e.g. to get
the Taylor expansion of an Eisenstein series
, or to evaluate
for positive integers .
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 |