# Book:Leonard Lewin/Polylogarithms and Associated Functions/Second Edition

## Leonard Lewin: Polylogarithms and Associated Functions (2nd Edition)

Published $1981$, North-Holland Publishing Co., New York-Amsterdam.

### BiBTeX

@book {MR618278,
AUTHOR = {Lewin, Leonard},
TITLE = {Polylogarithms and associated functions},
NOTE = {With a foreword by A. J. Van der Poorten},
PUBLISHER = {North-Holland Publishing Co., New York-Amsterdam},
YEAR = {1981},
PAGES = {xvii+359},
ISBN = {0-444-00550-1},
MRCLASS = {33A70 (10H99)},
MRNUMBER = {618278},
}

### Contents

Foreward
Preface
Preface to the Previous Work
CHAPTER 1. THE DILOGARITHM
1.1. Introduction and Elementary Considerations
$(1.1)$
$(1.2)$
$(1.3)$
---------
$(1.6)$
$(1.7)$
$(1.8)$
$(1.9)$
1.2. Extensions to Large Real Values of $z$
1.3. Functional Equations Involving a Single Variable
1.4. Numerical Relations
1.5. Functional Relations Involving Two Variables
1.6. Newman's Functional Equation
1.7. Functional Equations Involving Several Variables
1.8. Legendre's Chi-functions
1.9. Some Miscellaneous Results
1.10. A Survey of Definitions and Notations
1.11. Relations to Other Mathematical Functions
1.12. Occurrence in Physical Problems
CHAPTER 2. THE INVERSE TANGENT INTEGRAL
2.1. Elementary Considerations and Definition
2.2. The Inversion Relations
2.3. The Duplication Formula
2.4. Some Numerical Relations
2.5. The Triplication Formula
2.6. The Multiplication Formula for Odd Multiples
2.8. Functional Equations Involving Several Variables
CHAPTER 3. THE GENERALIZED INVERSE TANGENT INTEGRAL
3.1. Introduction and Elementary Properties
3.3. Differentiation with Respect to the Parameter
3.4. Formulas Arising from a Change of Variable
3.5. Formulas Arising form Inverse Tangent Integrals of Bilinear Argument
3.6. Factorization Theorems
3.7. Multiplication Theorems
3.8. Derived Relations
3.9. Special Values of the Parameter
3.10. An Addition Equation Involving Argument and Parameter
CHAPTER 4. CLAUSEN'S INTEGRAL
4.1 Definition and Elementary Properties
4.2 Periodic Properties
4.3 The Factorization Theorem
4.4 Series Expansions
4.5 Integral Relations
4.6 Functional Equations
4.7 Geometrical Connections
CHAPTER 5. THE DILOGARITHM OF COMPLEX ARGUMENT
5.1 Resolution into Real and Imaginary Parts
5.2 The Factorization Theorem
5.3 Special Values of the Argument
5.4 Functional Equations Involving a Single Variable
5.5 Reduction of $\mathrm{Li}_2(x,\theta)$ for Special Values of $\theta$
5.6 Newman's Functional Equation Involving Two Variables
5.7 Derived Functional Equations
5.8 Consequences of the Duplication Formula
5.9 An Addition Formula for the Angular Parameter
CHAPTER 6. THE TRILOGARITHM
6.1 Introduction and Elementary Considerations
6.2 Functional Equations of a Single Variable
6.3 Numerical Relations
6.4 A Consideration of Some Complex Forms
6.5 Functional Equations of Two Variables
6.6 A Further Consideration of Complex Forms
6.7 Functional Equations of Two Variables
6.8 A Functional Equation of Newman's Type
6.9 Functional Equations Involving Several Variables
CHAPTER 7. THE HIGHER-ORDER FUNCTIONS
7.1 Introduction and Definitions
7.2 The Inversion Equation and Its Consequences
7.3 The Factorization Theorem
7.4 Associated Integrals
7.5 The Associated Clausen Functions
7.6 Integral Relations for the Fourth-Order Polylogarithm
7.7 Functional Equations for the Fourth-Order Polylogarithm
7.8 Functional Equations for the Fifth-Order Polylogarithm
7.9 The Log-Sine Integrals
7.10 Results from a Contour Integration
7.11 Golden-Cut and Related Integrals
7.12 Polylogarithms of Nonintegral Order
7.13 Higher-Order Polylogarithms
CHAPTER 8. INTEGRATION OF FUNCTIONS AND SUMMATION OF SERIES
8.1 Reduction of a Class of Algebraic and Logarithmic Expressions
8.2 Reduction of Trigonometric Forms
8.3 Summation of Series
8.4 Integrals from the Higher-Order Functions
8.5 Definite Trigonometric Integrals
APPENDIX. REFERENCE DATA AND TABLES
A.1. Glossary of Notation
A.2. List of Selected Formulas
A.3. Reference List of Integrals
A.4. Tabulated Values
Bibliography
Suggestions for Further Study
Index