Partial fractions in complex analysis: Difference between revisions

Revision as of 10:25, 22 May 2006

Revision as of 02:40, 23 May 2006

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:<math>\tan(z) = \sum_{k=0}^{\infty} \frac{-2z}{z^2 - (k + \frac{1}{2})^2\pi^2}</math>

==Applications==

==Application to infinite products==

===Infinite products===

Because the partial fraction expansion often yields sums of ''1/(a+bz)'', it can be useful in finding a way to write a function as an [[infinite product]]; integrating both sides gives a sum of logarithms, and exponentiating gives the desired product:

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:<math>\cos z = \prod_{k=0}^{\infty} \left(1 - \frac{z^2}{(k + \frac{1}{2})^2\pi^2}\right).</math>

==Application to =Laurent series===

The partial fraction expansion for a function can also be used to find a Laurent series for it by simply replacing the rational functions in the sum with their Laurent series, which are often not difficult to write in closed form. This can also lead to interesting identities if a Laurent series is already known.