The Polylog function is defined as
Z(z,s) = z1/1s + z2/2s + z3/3s + ...It is equal to z L(z,1,s) for the more general Lerch transcendental
L(z,a,s) = z0/(0-a)s+ z1/(1-a)s + z2/(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 Lis(z) so that Lis(1) = Zeta(s) is the Riemann Zeta function. For s=2, one obtains the dilogarithm
Li2(z) = z/1 + z2/4 + z3/9 + ...For s=1, we have
Li1(z) = - log(1-z)For s=0 we have
Li0 = z/(1-z)The recursion z d/dz Lis+1(z) = = Lis(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)|