Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- 1 Formulation of physical problems
- 2 Classification of equations with two independent variables
- 3 One-dimensional waves
- 4 Finite domains and separation of variables
- 5 Elements of Fourier series
- 6 Introduction to Green's functions
- 7 Unbounded domains and Fourier transforms
- 8 Bessel functions and circular boundaries
- 9 Complex variables
- 10 Laplace transform and initial value problems
- 11 Conformal mapping and hydrodynamics
- 12 Riemann–Hilbert problems in hydrodynamics and elasticity
- 13 Perturbation methods – the art of approximation
- 14 Computer algebra for perturbation analysis
- Appendices
- Bibliography
- Index
8 - Bessel functions and circular boundaries
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- Acknowledgments
- 1 Formulation of physical problems
- 2 Classification of equations with two independent variables
- 3 One-dimensional waves
- 4 Finite domains and separation of variables
- 5 Elements of Fourier series
- 6 Introduction to Green's functions
- 7 Unbounded domains and Fourier transforms
- 8 Bessel functions and circular boundaries
- 9 Complex variables
- 10 Laplace transform and initial value problems
- 11 Conformal mapping and hydrodynamics
- 12 Riemann–Hilbert problems in hydrodynamics and elasticity
- 13 Perturbation methods – the art of approximation
- 14 Computer algebra for perturbation analysis
- Appendices
- Bibliography
- Index
Summary
Beyond elementary functions such as exponential, logarithmic, sinusoidal and hyperbolic functions, there is a host of so-called special functions that arise frequently in physical problems. Examples are Bessel functions, Legendre polynomials, Mathieu functions, hypergeometric functions, etc. Often these special functions emerge from the solution of partial differential equations when the boundary possesses a certain special geometry. For example, Bessel functions are associated with circular boundaries, while Legendre polynomials are associated with spherical boundaries, etc. In this chapter we choose to acquaint the readers only with the basic properties of the Bessel functions, and with applications in wave propagation and fluid flow. Certain essential facts such as series definitions, recursion formulas, orthogonality and asymptotic approximations will be discussed. Though far from exhaustive, these facts can already go a long way toward many applications, and can prepare the reader for further study of advanced aspects and other special functions. For quick access to further properties the reader should take advantage of some of the popular handbooks of special functions such as Erdelyi (1953a) and Abramowitz and Stegun (1964). For thorough theoretical expositions the reader must consult more advanced treatises such as Watson (1958).
Circular region and Bessel's equation
In this section we give a practical motivation for the need of Bessel functions by examining wave motion in a circular domain.
- Type
- Chapter
- Information
- Mathematical Analysis in EngineeringHow to Use the Basic Tools, pp. 165 - 209Publisher: Cambridge University PressPrint publication year: 1995