Partition function (number theory): Difference between revisions

Revision as of 02:25, 24 March 2005

Revision as of 02:43, 24 March 2005

Line 75:

{{inuse}}

==Rademacher's series==

InAn [[1937asymptotic]], [[Hans Rademacher]] was able to obtain a convergent series expression for ''p''(''kn''). is given by

:<math>p(n) \sim \frac {\exp \pi \sqrt {2n/3}} {4n\sqrt{3}} \mbox { as } n\rightarrow \infty</math>

This expression was first optained by [[G. H. Hardy]] and [[Ramanujan]] in [[1918]] and independently by [[J. V. Uspensky]] in [[1920]].

In [[1937]], [[Hans Rademacher]] was able to improve on Hardy and Ramanjan's results by providing a convergent series expression for ''p''(''n''). It is

:<math>p(n)=\frac{1}{\pi \sqrt{2}} \sum_{k=1}^\infty A_k(n)\;

\sqrt{k} \; \frac{d}{dn}

\left( \frac {\sinh \left( \frac{\pi}{k}

\sqrt{\frac{2}{3}\left(n-\frac{1}{24}\right)}\right) }

{\sqrt{n-\frac{1}{24}}}\right)

</math>

where

==Refernces==