# Book:Leonard Lewin/Dilogarithms and Associated Functions

## Leonard Lewin: Dilogarithms and Associated Functions

Published $1958$, Macdonald & Co. Ltd..

### Contents

Foreward
Preface
Chapter I The Dilogarithm
1. Introduction and Elementary Considerations
$(1.1)$
$(1.2)$
2. Extension to Large Real Values of $z$
3. Functional Equations Involving a Single Variable
4. Numerical Relations
5. Functional Relations Involving Two Variables
6. Newman's Functional Equation
7. Functional Equations Involving Several Variables
8. Legendre's Chi-function
9. Some Miscellaneous Results
10. A Survey of Definitions and Notations
11. Relations to Other Mathematical Functions
12. Occurrence in Physical Problems
Chapter II The Inverse Tangent Integral
1. Elementary Considerations and Definitions
2. The Inversion Relation
3. The Duplication Formula
4. Some Numerical Relations
5. The Triplication Formula
6. The Multiplication Formula for Odd Multiples
7. The Quadruplication Formula
8. Function Equations Involving Several Variables
Chapter III The Generalised Inverse Tangent Integral
1. Introduction and Elementary Properties
2. Differentiation with Respect to the Parameter
3. Formulae Arising from a Change of Variable
4. Formulae Arising from Inverse Tangent Integrals of Bi-Linear Argument
5. Formulae Arising from Inverse Tangent Integrals of Bi-Quadratic Argument
6. Factorisation Theorems
7. Multiplication Formulae
8. Derived Relations
9. Special Values of the Parameter
10. An Addition Equation Involving Argument and Parameter
Chapter IV Clausen's Integrals
1. Definition and Elementary Properties
2. Periodic Properties
3. The Factorisation Theorem
4. Series Expansions
5. Integral Relations
6. Functional Equations
Chapter V The Dilogarithm of Complex Argument
1. Resolution into Real and Imaginary Parts
2. The Factorisation Theorem
3. Special Values of the Argument
4. Functional Equations Involving a Single Variables
5. Defuction of $\mathrm{Li}_2(x,\theta)$ for Special Values of $\theta$
6. Newman's Functional Equation Involving Two Variables
7. Derived Functional Equations
8. Consequences of the Duplication Formula
9. An Addition Formula for the Angular Parameter
Chapter VI The Trilogarithm
1. Introduction and Elementary Considerations
2. Functional Equations of a Single Variable
3. Numerical Relations
4. A Consideration of Some Complex Forms
5. A Generalisation of Clausen's Integral
6. A Further Consideration of Complex Forms
7. Functional Equations of Two Variables
8. A Functional Equation of Newman's Type
9. Functional Equations Involving Several Variables
Chapter VII The Higher Order Functions
1. Introduction and Definitions
2. The Inversion Equation and Its Consequences
3. The Factorisation Theorem
4. Associated Integrals
5. The Associated Clausen Functions
6. Integral Relations for the Fourth Order Polylogarithm
7. Functional Equations for the Fourth Order Polylogarithm
8. Functional Equations for the Fifth Order Polylogarithm
9. The Log-Sine Integrals
10. Higher Order Polylogarithms
Chapter VIII The Integration of Functions and Summation of Series
1. The Reduction of a Class of Algebraic and Logarithmic Expressions
2. The Reduction of Trigonometric Forms
3. Summation of Series
4. Integrals from the Higher Order Functions
Chapter IX Reference Data and Tables
1. Glossary of Notation
2. List of Selected Formulae
3. Reference List of Integrals
4. Tabulated Values
5. Bibliography
6. Suggestions for Further Study