Polylogarithm

Abstract: Hilbert and Polya suggested that there might be a natural spectral interpretation of the zeroes of the Riemann Zeta function. While at the time there was little evidence for this, today the evidence is quite convincing. Firstly, there are the “function field” analogues, that is zeta functions of curves over finite fields and their generalizations. For these a spectral interpretation for their zeroes exists in terms of eigenvalues of Frobenius on cohomology. Secondly, the developments, both theoretical and numerical, on the local spacing distributions between the high zeroes of the zeta function and its generalizations give striking evidence for such a spectral connection. Moreover, the low-lying zeroes of various families of zeta functions follow laws for the eigenvalue distributions of members of the classical groups. In this paper we review these developments. In order to present the material fluently, we do not proceed in chronological order of discovery. Also, in concentrating entirely on the subject matter of the title, we are ignoring the standard body of important work that has been done on the zeta function and L-functions. 1. The Montgomery-Odlyzko Law We begin with the Riemann Zeta function and some phenomenology associated with it.

Abstract: We present a review of the symbol map, a mathematical tool that can be useful in simplifying expressions among multiple polylogarithms, and recall its main properties. A recipe is given for how to obtain the symbol of a multiple polylogarithm in terms of the combinatorial properties of an associated rooted decorated polygon. We also outline a systematic approach to constructing a function corresponding to a given symbol, and illustrate it in the particular case of harmonic polylogarithms up to weight four. Furthermore, part of the ambiguity of this process is highlighted by exhibiting a family of non-trivial elements in the kernel of the symbol map for arbitrary weight.

Abstract: GENERALIZED GAMMA FUNCTION The Gamma Fun ction G(a) Definition of the Generalized Gamma Function Properties of the Generalized Gamma Function Mellin and Laplace Transforms Asymptotic Representations The Macdonald Function The Digamma Function y(x) Generalization of Psi (Digamma) Function Integral Representations of yb (a) Properties of the Generalized Psi Function Graphical and Tabular Representations THE GENERALIZED INCOMPLETE GAMMA FUNCTIONS The Incomplete Gamma Functions Definition of the Generalized Incomplete Gamma Functions Properties of the Incomplete Generalized Gamma Functions Convolution Representations and Laplace Transforms Connection with Other Special Functions KdF Functions and Incomplete Integrals Representation in terms of KdF Functions Reduction Formulas for F0:2 1 2:0 0[x,y] Integrals of Product of Bessel and Gamma Functions Asymptotic Representations Integral Representation for G(a,x b) Graphical and Tabular Representations THE FAMILY OF THE GAMMA FUNCTIONS The Family of Incomplete Gamma Functions The Generalized Error Functions The Generalized Exponential Integral function The Generalized Fresnel Integrals The Decomposition Functions The Extended Decomposition functions The E(u,v) and F(u,v) Functions The e(u) and f(u) Functions Graphical and Tabular Representations EXTENSION OF GENERALIZED INCOMPLETE GAMMA FUNCTIONS Introduction The Decomposition Formula Recurrence Relations Laplace and K-Transform Representation Parametric Differentiation and Integration Connection with Other Special Functions Integral Representations Differential Representations The Mellin Transform Representation EXTENDED BETA FUNCTION The Beta Function The Incomplete Beta Function The Beta Probability Distribution Definition of the Extended Beta Function Properties of the Extended Beta Function Integral Representations of the Extended Beta Function Conncection with Other Special Functions Representations in Terms of Whittaker functions Extended Incomplete Beta Function The Extended Beta Distribution Graphical and Tabular Representations EXTENDED INCOMPLETE GAMMA FUNCTIONS Introduction Definition of the Extended Incomplete Gamma Functions The Decomposition Formula Recurrence Formula Connection with Other Special Functions The H-Function Incomplete Fox H-Functions EXTENDED RIEMANN ZETA FUNCTIONS Introduction Bernoulli's Numbers and Polynomials The Zeta Function Zeros of the Zeta Function and the Function p(x) The Extended Zeta Function z(a) The Second Extended Zeta Function zb*(a) The Hurwitz Zeta Function Extended Hurwitz Zeta Functions Extended Hurwitz Formulae Further Remarks and Comments Graphical and Tabular Representations PHASE-CHANGE HEAT-TRANSFER Introduction Constant Temperature Boundary Conditions Convective Boundary Conditions Freezing of Tissues around a Capillary Tube Freezing of Binary Alloys Freezing Around an Impurity Numerical Methods for Phase-Change Problems HEAT CONDUCTION WITH TIME-DEPENDENT BOUNDARY CONDITIONS Introduction Time-Dependent Surface Temperatures Time-Dependent Surface Heat Fluxes Illustrative Example HEAT CONDUCTION DUE TO TIME-DEPENDENT LASER SOURCES Introduction Mathematical Formulation Some Cases of Practical Interest A UNIFIED APPROACH TO HEAT SOURCE PROBLEMS Introduction Thermal Explosions Continuously Operating Heat Sources APPENDICES Heat Conduction Table of Laplace Transforms Integrals Dependent of Parameters REFERENCES SYMBOLS INDEX