The Polylog function is defined as
Z(z,s) = z^{1}/1^{s} + z^{2}/2^{s} + z^{3}/3^{s} + ...It is equal to z L(z,1,s) for the more general Lerch transcendental L(z,a,s) = z^{0}/(0-a)^{s}+ z^{1}/(1-a)^{s} + z^{2}/(2-a)^{s} + ...For fixed z different from 1, the Lerch transcendental and so the polylog function have analytic continuations to the entire s-plane. One often looks at the polylog function as a function of z. In this case, when s is a parameter, one denotes the function by Li_{s}(z) so that Li_{s}(1) = Zeta(s) is the Riemann Zeta function. For s=2, one obtains the dilogarithm Li_{2}(z) = z/1 + z^{2}/4 + z^{3}/9 + ...For s=1, we have Li_{1}(z) = - log(1-z)For s=0 we have Li_{0} = z/(1-z)The recursion z d/dz Li_{s+1}(z) = = Li_{s}(z) shows that for negative integer values of s, the polylog function is a rational function in z which can be computed in closed form. |
A plot of the Lerch function for a=1 (polylog), alpha = golden mean. More precisely the plot shows level curves of the function f(x,y) = |Z(exp(2pi i (sqrt(5)-1)/2),x+i y)| |