Polylogarithms and associated functions

Title Page Table of Contents Foreword Preface Preface to the Previous Work CHAPTER 1. THE DILOGARITHM 1.1. Introduction and Elementary Considerations 1.2. Extension 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-function 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 Relation 2.3. The Duplication Formula 2.4. Some Numerical Relations 2.5. The Triplication Formula 2.6. The Multiplication Formula for Odd Multiples 2.7. The Quadruplication Formula 2.8. Functional Equations Involving Several Variables CHAPTER 3. THE GENERALIZED INVERSE TANGENT INTEGRAL 3.1. Introduction and Elementary Properties 3.2. Differentiation with Respect to the Parameter 3.3. Formulas Arising from a Change of Variable 3.4. Formulas Arising from Inverse Tangent Integrals of Bilinear Argument 3.5. Formulas Arising from Inverse Tangent Integrals of Biquadratic Argument 3.6. Factorization Theorems 3.7. Multiplication Formulas 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 Li[2](x,theta) for Special Values of theta 5.6. Newman’s FunctionaI 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. A Generalization of Clausen’s Integral 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 Theorems 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