Explicit evaluation of some integrals involving polylogarithm functions

Abstract

In this paper, we give explicit evaluation for some integrals involving polylogarithm functions of types ${\int }_0^x{t}^{m}L{i}_p\left(t\right)\mathrm{d}t$ and ${\int }_0^x{\log }^m\left(t\right)L{i}_p\left(t\right)\mathrm{d}t$. Some more integrals involving the logarithm function will also be derived.

Keywords:

• Polylogarithm function
• recurrence relation
• zeta function

2010 MR Subject Classifications:

• 33E20
• 11B37
• 11M06

Acknowledgements

The author is grateful to the referee for her/his useful comments and suggestions. The author is also grateful to Dr. Wanfeng Liang and Dr. Ke Wang for some useful discussions.

Disclosure statement

No potential conflict of interest was reported by the author