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A Class of Logarithmic Integrals Victor Adamchik

Summary: A Class of Logarithmic Integrals
Victor Adamchik
Wolfram Research Inc.
100 Trade Center Dr.
Champaign, IL 61820, USA
April 10, 1997
Abstract. A class of definite integrals involving cyclotomic polynomials and nested
logarithms is considered. The results are given in terms of derivatives of the Hur­
witz Zeta function. Some special cases for which such derivatives can be expressed
in closed form are also considered. The integration procedure is implemented in
Mathematica V3.1.
1 Introduction
The aim of the paper is to develop an approach for evaluating a class of integrals
involved cyclotomic polynomials and the nested logarithms log log x. This class of
integrals arose from the research regarding the Potts model on the triangular lattice
(see [1], [2]). The Potts model encompasses a number of problems in statistical
physics and lattice theory. It generalizes the Ising model so that each spin can have
more than two values. It includes the ice­vertex and bond percolation models as
special cases. It is also related to graph­coloring problems. Baxter, Temperley and
Ashley (see [3]) derived the following generating function for the Potts model on the


Source: Adamchik, Victor - School of Computer Science, Carnegie Mellon University


Collections: Computer Technologies and Information Sciences