This volume studies classical and modern special functions arising as solutions to differential equations, integral transforms, and representation...
This volume studies classical and modern special functions arising as solutions to differential equations, integral transforms, and representation theory. It emphasizes structure, identities, and computational aspects.
Part I. Foundations
Chapter 1. What Are Special Functions
1.1 Definition and scope 1.2 Historical origins 1.3 Functions as solutions to equations 1.4 Orthogonality and completeness 1.5 Examples
Chapter 2. Series and Integral Representations
2.1 Power series definitions 2.2 Integral representations 2.3 Generating functions 2.4 Recurrence relations 2.5 Asymptotic behavior
Chapter 3. Differential Equations
3.1 Linear differential equations 3.2 Singular points 3.3 Frobenius method 3.4 Classification of equations 3.5 Examples
Part II. Classical Functions
Chapter 4. Gamma and Beta Functions
4.1 Definitions 4.2 Functional equations 4.3 Integral representations 4.4 Properties 4.5 Applications
Chapter 5. Bessel Functions
5.1 Bessel differential equation 5.2 Series solutions 5.3 Orthogonality 5.4 Recurrence relations 5.5 Applications
Chapter 6. Legendre and Related Functions
6.1 Legendre polynomials 6.2 Associated functions 6.3 Orthogonality 6.4 Generating functions 6.5 Applications
Part III. Orthogonal Polynomials
Chapter 7. Orthogonal Systems
7.1 Inner products 7.2 Orthogonality relations 7.3 Weight functions 7.4 Completeness 7.5 Examples
Chapter 8. Classical Families
8.1 Hermite polynomials 8.2 Laguerre polynomials 8.3 Chebyshev polynomials 8.4 Properties 8.5 Applications
Chapter 9. General Theory
9.1 Recurrence relations 9.2 Rodrigues formulas 9.3 Zeros and distribution 9.4 Approximation properties 9.5 Applications
Part IV. Hypergeometric Functions
Chapter 10. Hypergeometric Series
10.1 Definitions 10.2 Convergence 10.3 Differential equations 10.4 Transformation formulas 10.5 Examples
Chapter 11. Generalized Hypergeometric Functions
11.1 Definitions 11.2 Properties 11.3 Relations 11.4 Applications 11.5 Examples
Chapter 12. Special Cases and Identities
12.1 Classical identities 12.2 Transformation theory 12.3 Summation formulas 12.4 Applications 12.5 Examples
Part V. Asymptotics and Approximation
Chapter 13. Asymptotic Expansions
13.1 Definitions 13.2 Methods 13.3 Applications 13.4 Examples 13.5 Error estimates
Chapter 14. Approximation Methods
14.1 Series truncation 14.2 Numerical approximation 14.3 Stability 14.4 Applications 14.5 Examples
Chapter 15. Integral Methods
15.1 Laplace method 15.2 Stationary phase 15.3 Saddle-point methods 15.4 Applications 15.5 Examples
Part VI. Transform Methods
Chapter 16. Fourier Transform
16.1 Definitions 16.2 Properties 16.3 Relation to special functions 16.4 Applications 16.5 Examples
Chapter 17. Laplace Transform
17.1 Definitions 17.2 Inversion 17.3 Differential equations 17.4 Applications 17.5 Examples
Chapter 18. Mellin Transform
18.1 Definitions 18.2 Properties 18.3 Applications 18.4 Examples 18.5 Connections
Part VII. Applications
Chapter 19. Physics Applications
19.1 Quantum mechanics 19.2 Wave equations 19.3 Heat equation 19.4 Applications 19.5 Examples
Chapter 20. Engineering Applications
20.1 Signal processing 20.2 Control systems 20.3 Electromagnetics 20.4 Applications 20.5 Examples
Chapter 21. Computational Aspects
21.1 Numerical evaluation 21.2 Stability issues 21.3 Software libraries 21.4 Symbolic computation 21.5 Applications
Part VIII. Research Directions
Chapter 22. Advanced Topics
22.1 q-series and basic hypergeometric functions 22.2 Special functions in representation theory 22.3 Orthogonal polynomials in several variables 22.4 Modern developments 22.5 Emerging areas
Chapter 23. Open Problems
23.1 Functional identities 23.2 Asymptotic behavior 23.3 Computational challenges 23.4 Connections to other fields 23.5 Future directions
Chapter 24. Historical and Conceptual Notes
24.1 Development of special functions 24.2 Key contributors 24.3 Evolution of methods 24.4 Cross-disciplinary impact 24.5 Summary
Appendix
A. Common formulas and identities B. Orthogonality tables C. Proof techniques checklist D. Example catalog E. Cross-reference to other MSC branches
This volume develops special functions as structured solutions to mathematical and physical problems. It emphasizes their unifying role across analysis, physics, and computation.