Engineering Mathematics: Feb 11, 2013

Definition of Polylogarithm

$Li_{s}(z) = \sum _{k=1}^{\infty }\frac{z^{k}}{k^{s}}$

PL.ID.1: Functional Equation of Polylogarithm

$Li_{s}(z) + Li_{s}(-z) = 2^{1-s}Li_{s}(z^{2})$

Proof

$Li_{s}(z) = \sum _{k=1}^{\infty }\frac{z^{k}}{k^{s}}= z+ \frac{z^{2}}{2^{s}}+ \frac{z^{3}}{3^{s}} +\cdot \cdot \cdot$

(1)

$Li_{s}(-z) = \sum _{k=1}^{\infty }\frac{(-z)^{k}}{k^{s}}= -z+ \frac{z^{2}}{2^{s}}- \frac{z^{3}}{3^{s}} \pm \cdot \cdot \cdot$

(2)

summing (1) and (2)

$Li_{s}(z) + Li_{s}(-z) = (z-z)+ (\frac{z^{2}}{2^{s}}+\frac{z^{2}}{2^{s}})+= (\frac{z^{3}}{3^{s}} -\frac{z^{3}}{3^{s}} )+ \cdot \cdot \cdot = 2(\frac{z^{2}}{2^{s}} +\frac{z^{4}}{4^{s}} +\cdot \cdot \cdot)$

$Li_{s}(z) + Li_{s}(-z) = 2\sum _{k=2,4,6\cdot \cdot \cdot }^{\infty }\frac{z^{k}}{k^{s}}$

(3)