Polylogarithm

ayush007

this may prove quite interesting for you all........

the reference may give you an idea of how really a non integrable looking function may actually be integrable....

Polylogarithm

From Wikipedia, the free encyclopedia

The polylogarithm (also known as de Jonquière's function) is a special function Li_s(z) that is defined by the sum

$\operatorname{Li}_s(z) = \sum_{k=1}^\infty {z^k \over k^s}.$

The above definition is valid for all complex numbers s and z where |z|< 1. The polylogarithm is defined over a larger range of z than the above definition allows by the process of analytic continuation.

The special case s = 1 involves the ordinary natural logarithm (Li₁(z)=-ln(1-z)) while the special cases s = 2 and s = 3 are called the dilogarithm (also referred to as Spence's function) and trilogarithm respectively. The name of the function comes from the fact that it may alternatively be defined as the repeated integral of itself, namely that

$\operatorname{Li}_{s+1}(z) = \int_0^z \frac {\operatorname{Li}_s(t)}{t}dt$

so that the dilogarithm is an integral of the logarithm, and so on. For negative integer values of s, the polylogarithm is a rational function.

The polylogarithm also arises in the closed form of the integral of the Fermi-Dirac distribution and the Bose-Einstein distribution and is sometimes known as the Fermi-Dirac integral or the Bose-Einstein integral. Polylogarithms should not be confused with polylogarithmic functions nor with the offset logarithmic integral which has a similar notation.

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Polylogarithm

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