Meromorphic functions connected with polylogarithms (Q1814361)

The polylogarithm \[ Li_ m(z)= \sum_{n=1}^ \infty z^ n/n^ m \] cannot be extended to a continuous one-valued function on the whole complex plane. There are a number of suggested remedies: for example, \textit{D. Zagier} [``Special values and functional equations of polylogarithms'', in L. Lewin (ed.), ``Structural properties of polylogarithms'' (1991; Zbl 0745.33009)] works with \[ P_ m(z)={\mathfrak R}_ m\left( \sum_{i=0}^{m-1} {{2^ j B_ j} \over {j!}} (\log | z|)^ j Li_{m_ j}(z)\right) \] where \({\mathfrak R}_ m\) denotes the imaginary or the real part according as \(m\) is even or odd. This is continuous real-valued and real analytic except at 0, 1 and infinity. The present paper contains a similar construction and an unmotivated account of matters related to the proof.