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The Lerch $Φ$ Analytic Continuation

Jose Risomar Sousa

Abstract

We demonstrate how to extend formulae for the Lerch transcendent function, $Φ(e^z,k,b)$, and the polylogarithm, $\mathrm{Li}_{k}(e^{z})$, that only hold at the positive integers to the right half of the complex $k$-plane, that is, $\Re{(k)}>0$. The same is done for the partial sums of each of these functions.

The Lerch $Φ$ Analytic Continuation

Abstract

We demonstrate how to extend formulae for the Lerch transcendent function, , and the polylogarithm, , that only hold at the positive integers to the right half of the complex -plane, that is, . The same is done for the partial sums of each of these functions.

Paper Structure

This paper contains 19 sections, 39 equations.

Table of Contents

  1. Introduction
  2. The formulae at the positive integers
  3. Partial Lerch $\Phi$
  4. Full Lerch $\Phi$
  5. Non-integer $2b$
  6. Half-integer $b$
  7. Integer $b$
  8. Partial polylogarithm
  9. Full polylogarithm
  10. Polylogarithm extension
  11. Partial polylogarithm
  12. $H_k(n)$ when $\Re{(k)}>-1$
  13. $\zeta(k)$ outside the critical strip
  14. Full polylogarithm
  15. Lerch $\Phi$ extension
  16. ...and 4 more sections