Special functions [[electronic resource] ] : an introduction to the classical functions of mathematical physics / / Nico M. Temme

• Autore: Temme N. M
• Pubblicazione: New York, : Wiley, 1996
• Descrizione fisica: 1 online resource (392 p.)
• Disciplina: 515.5; 530.15
• Soggetto topico: Functions, Special; Boundary value problems; Mathematical physics
• Note generali: "A Wiley-Interscience publication"--t.p.
• Nota di bibliografia: Includes bibliographical references (p. 349-360) and index.
• Nota di contenuto: Special Functions: An Introduction to the Classical Functions of Mathematical Physics; Contents; 1 Bernoulli, Euler and Stirling Numbers; 1.1. Bernoulli Numbers and Polynomials; 1.1.1. Definitions and Properties; 1.1.2. A Simple Difference Equation; 1.1.3. Euler's Summation Formula; 1.2. Euler Numbers and Polynomials; 1.2.1. Definitions and Properties; 1.2.2. Boole's Summation Formula; 1.3. Stirling Numbers; 1.4. Remarks and Comments for Further Reading; 1.5. Exercises and Further Examples; 2 Useful Methods and Techniques; 2.1. Some Theorems from Analysis ; 2.2. Asymptotic Expansions of Integrals2.2.1. Watson's Lemma; 2.2.2. The Saddle Point Method; 2.2.3. Other Asymptotic Methods; 2.3. Exercises and Further Examples; 3 The Gamma Function; 3.1. Introduction; 3.1.1. The Fundamental Recursion Property; 3.1.2. Another Look at the Gamma Function; 3.2. Important Properties; 3.2.1. Prym's Decomposition; 3.2.2. The Cauchy-Saalschütz Representation; 3.2.3. The Beta Integral; 3.2.4. The Multiplication Formula; 3.2.5. The Reflection Formula; 3.2.6. The Reciprocal Gamma Function; 3.2.7. A Complex Contour for the Beta Integral; 3.3. Infinite Products ; 3.3.1. Gauss' Multiplication Formula3.4. Logarithmic Derivative of the Gamma Function; 3.5. Riemann's Zeta Function; 3.6. Asymptotic Expansions; 3.6.1. Estimations of the Remainder; 3.6.2. Ratio of Two Gamma Functions; 3.6.3. Application of the Saddle Point Method; 3.7. Remarks and Comments for Further Reading; 3.8. Exercises and Further Examples; 4 Differential Equations; 4.1. Separating the Wave Equation; 4.1.1. Separating the Variables; 4.2. Differential Equations in the Complex Plane; 4.2.1. Singular Points; 4.2.2. Transformation of the Point at Infinity ; 4.2.3. The Solution Near a Regular Point4.2.4. Power Series Expansions Around a Regular Point; 4.2.5. Power Series Expansions Around a Regular Singular Point; 4.3. Sturm's Comparison Theorem; 4.4. Integrals as Solutions of Differential Equations; 4.5. The Liouville Transformation; 4.6. Remarks and Comments for Further Reading; 4.7. Exercises and Further Examples; 5 Hypergeometric Functions; 5.1. Definitions and Simple Relations; 5.2. Analytic Continuation; 5.2.1. Three Functional Relations; 5.2.2. A Contour Integral Representation; 5.3. The Hypergeometric Differential Equation ; 5.4. The Singular Points of the Differential Equation5.5. The Riemann-Papperitz Equation; 5.6. Barnes' Contour Integral for F(a, b; c; z); 5.7. Recurrence Relations; 5.8. Quadratic Transformations; 5.9. Generalized Hypergeometric Functions; 5.9.1. A First Introduction to q-functions; 5.10. Remarks and Comments for Further Reading; 5.11. Exercises and Further Examples; 6 Orthogonal Polynomials; 6.1. General Orthogonal Polynomials; 6.1.1. Zeros of Orthogonal Polynomials; 6.1.2. Gauss Quadrature; 6.2. Classical Orthogonal Polynomials; 6.3. Definitions by the Rodrigues Formula ; 6.4. Recurrence Relations
• Sommario/riassunto: This book gives an introduction to the classical, well-known special functions which play a role in mathematical physics, especially in boundary value problems. Calculus and complex function theory form the basis of the book and numerous formulas are given. Particular attention is given to asymptomatic and numerical aspects of special functions, with numerous references to recent literature provided.
• Titolo autorizzato: Special functions
• ISBN: 1-280-76794-4; 9786613678713; 1-118-03257-8; 1-118-03081-8
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910841474403321
• Lo trovi qui: Univ. Federico II