Polylogarithm

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High Quality Content by WIKIPEDIA articles! In the important case where the parameter s is an integer, it will be represented by n...

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High Quality Content by WIKIPEDIA articles! In the important case where the parameter s is an integer, it will be represented by n (or n when negative). It is often convenient to define = ln(z) where ln(z) is the principal branch of the complex logarithm Ln(z) so that Im( ) . Also, all exponentiation will be assumed to be single valued: zs = exp(s ln(z)). Depending on the parameter s, the polylogarithm may be multi-valued. The principal branch of the polylogarithm is chosen to be that for which Lis(z) is real for z real, 0 z 1 and is continuous except on the positive real axis, where a cut is made from z = 1 to such that the cut puts the real axis on the lower half plane of z. In terms of , this amounts to arg( ) . The fact that the polylogarithm may be discontinuous in can cause some confusion.