The function

(1) |

(2) |

The polylogarithm satisfies the fundamental identities

(3) |

(4) |

(5) |

(6) |

The polylogarithm identities lead to remarkable expressions. Ramanujan gave the polylogarithm identities

(7) |

(8) |

(9) |

(10) |

(11) |

(12) |

(13) |

(14) |

(15) |

(16) |

(17) |

No general Algorithm is know for the integration of polylogarithms of functions.

**References**

Bailey, D.; Borwein, P.; and Plouffe, S. ``On the Rapid Computation of Various Polylogarithmic Constants.'' http://www.cecm.sfu.ca/~pborwein/PAPERS/P123.ps.

Berndt, B. C. *Ramanujan's Notebooks, Part IV.* New York: Springer-Verlag, pp. 323-326, 1994.

Lewin, L. *Polylogarithms and Associated Functions.* New York: North-Holland, 1981.

Lewin, L. (Ed.). *Structural Properties of Polylogarithms.* Providence, RI: Amer. Math. Soc., 1991.

Nielsen, N. *Der Euler'sche Dilogarithms.* Leipzig, Germany: Halle, 1909.

© 1996-9

1999-05-25