# Book:Leonard Lewin/Dilogarithms and Associated Functions

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## Leonard Lewin: *Dilogarithms and Associated Functions*

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

### Contents

- Foreward
- Preface
- Chapter I The Dilogarithm
- 1. Introduction and Elementary Considerations
- 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