Question: partition function

My colleague at the university next week give a lecture where he tells about the Hardy-Ramanujan partition functions, the usual

notations are p(n) and q(n). He asked me, could I calculate by Maple a definite integral, because the answer was yes, then he

would say "by Maple".  I tried it in different ways, but no success. Could someone calculate it?

The exact result is  -1/2 ln (2 Pi) .

[;\int_{0}^{\infty} \frac{\left(\frac{1}{\exp(x)-1}-\frac{1}{x}+\frac{\exp(-x)}{2}\right)}{x} dx=?;]?

Any idea are welcome. Thanks,


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