Polylogarithm

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Bol 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.
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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.

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