Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 14E (Further Complex Methods)

For fixed $s$, the factor $n^{-s}$ grows at most polynomially in $n$, whereas $|z{|}^n$ decays exponentially when $|z|<1$. The defining series for the polylogarithm therefore converges absolutely and locally uniformly for every complex $s$ and $|z|<1$. It is holomorphic in $z$ on the unit disc and entire in $s$ there.

For $|z|<1$ on the negative-real portions of the Hankel contour, expandLocal uniform convergence permits termwise integration. The reciprocal Hankel formula givesso $I(z,s)={\sum }_{n\ge 1}{z}^n/n^s={\mathrm{Li}}_s(z)$ initially on the disc.

For $z\notin [1,\infty )$, choose the contour around the negative axis so that it passes between zero and every pole satisfying $e^{-t}=z$. On compact subsets of the slit $z$-plane it can be chosen uniformly, and differentiation under the integral proves holomorphy. Deforming without crossing a pole gives a single-valued analytic continuation. Near the cut, the pole $t=-\log z$ lies on opposite sides of the two admissible contours; this is why the contour must pass it consistently.

When $s$ is a nonpositive integer, $t^{s-1}$ is meromorphic and the Hankel contour may be shrunk to a small circle about zero. The integral is then computed by the residue at zero. This givesThese also follow by repeatedly applying $zd/dz$ to the geometric series.

For real $s>0$, collapse the Hankel contour onto the negative axis and use the jump in the argument of $t^{s-1}$. The reflection formula for the gamma function yieldsso $K(z,s,t)=z{t}^{s-1}/[\Gamma (s)(e^t-z)]$. At positive integer $s$, the original Hankel expression is interpreted by analytic continuation: the pole of $\Gamma (1-s)$ cancels the corresponding zero of the contour integral.

When $z$ crosses the slit at $x>1$, the pole of the real-integral kernel at $t=\log x$ crosses the integration path. The two boundary values differ by $2\pi i$ times its residue. Since $d(e^t-z)/dt=x$ there, that residue is $(\log x{)}^{s-1}/\Gamma (s)$, giving the jump