Chapter 10 Bessel Functions

F. W. J. Olver Institute for Physical Science and Technology and Department of Mathematics, University of Maryland, College Park, Maryland. L. C. Maximon Center for Nuclear Studies, Department of Physics, The George Washington University, Washington, D.C.

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Acknowledgements:

This chapter is based in part on Abramowitz and Stegun (1964, Chapters 9, 10, and 11) by F. W. J. Olver, H. A. Antosiewicz, and Y. L. Luke, respectively. The authors are pleased to acknowledge assistance of Martin E. Muldoon with §§10.21 and 10.42, Adri Olde Daalhuis with the verification of Eqs. (10.15.6)–(10.15.9), (10.38.6), (10.38.7), (10.60.7)–(10.60.9), and (10.61.9)–(10.61.12), Peter Paule and Frédéric Chyzak for the verification of Eqs. (10.15.6)–(10.15.9), (10.38.6), (10.38.7), (10.56.1)–(10.56.5), (10.60.4), (10.60.6), (10.60.10), and (10.60.11) by application of computer algebra, Nico Temme with the verification of Eqs. (10.15.6)–(10.15.9), (10.38.6), and (10.38.7), and Roderick Wong with the verification of §§10.22(v) and 10.43(v).

Notes:

The main references used in writing this chapter are Watson (1944) and Olver (1997b).

Referenced by:

§3.10(ii), New Associate Editors for the DLMF, Profile Diego Dominici, Profile Frank W. J. Olver, Profile Gergő Nemes, Profile Leonard C. Maximon, § ‣ Chapter Authors, § ‣ Chapter Authors, § ‣ Associate Editors, § ‣ Associate Editors, Erratum (V1.1.6) for Chapters 10 Bessel Functions, 18 Orthogonal Polynomials, 34 3j, 6j, 9j Symbols, Erratum (V1.1.6) for Chapters 1 Algebraic and Analytic Methods, 10 Bessel Functions, 14 Legendre and Related Functions, 18 Orthogonal Polynomials, 29 Lamé Functions, Version 1.2.0 (March 27, 2024), § ‣ Software Cross Index

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• Notation
  1. 10.1 Special Notation
• Bessel and Hankel Functions
  1. 10.2 Definitions
  2. 10.3 Graphics
  3. 10.4 Connection Formulas
  4. 10.5 Wronskians and Cross-Products
  5. 10.6 Recurrence Relations and Derivatives
  6. 10.7 Limiting Forms
  7. 10.8 Power Series
  8. 10.9 Integral Representations
  9. 10.10 Continued Fractions
  10. 10.11 Analytic Continuation
  11. 10.12 Generating Function and Associated Series
  12. 10.13 Other Differential Equations
  13. 10.14 Inequalities; Monotonicity
  14. 10.15 Derivatives with Respect to Order
  15. 10.16 Relations to Other Functions
  16. 10.17 Asymptotic Expansions for Large Argument
  17. 10.18 Modulus and Phase Functions
  18. 10.19 Asymptotic Expansions for Large Order
  19. 10.20 Uniform Asymptotic Expansions for Large Order
  20. 10.21 Zeros
  21. 10.22 Integrals
  22. 10.23 Sums
  23. 10.24 Functions of Imaginary Order
• Modified Bessel Functions
  1. 10.25 Definitions
  2. 10.26 Graphics
  3. 10.27 Connection Formulas
  4. 10.28 Wronskians and Cross-Products
  5. 10.29 Recurrence Relations and Derivatives
  6. 10.30 Limiting Forms
  7. 10.31 Power Series
  8. 10.32 Integral Representations
  9. 10.33 Continued Fractions
  10. 10.34 Analytic Continuation
  11. 10.35 Generating Function and Associated Series
  12. 10.36 Other Differential Equations
  13. 10.37 Inequalities; Monotonicity
  14. 10.38 Derivatives with Respect to Order
  15. 10.39 Relations to Other Functions
  16. 10.40 Asymptotic Expansions for Large Argument
  17. 10.41 Asymptotic Expansions for Large Order
  18. 10.42 Zeros
  19. 10.43 Integrals
  20. 10.44 Sums
  21. 10.45 Functions of Imaginary Order
  22. 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function

• Spherical Bessel Functions
  1. 10.47 Definitions and Basic Properties
  2. 10.48 Graphs
  3. 10.49 Explicit Formulas
  4. 10.50 Wronskians and Cross-Products
  5. 10.51 Recurrence Relations and Derivatives
  6. 10.52 Limiting Forms
  7. 10.53 Power Series
  8. 10.54 Integral Representations
  9. 10.55 Continued Fractions
  10. 10.56 Generating Functions
  11. 10.57 Uniform Asymptotic Expansions for Large Order
  12. 10.58 Zeros
  13. 10.59 Integrals
  14. 10.60 Sums
• Kelvin Functions
  1. 10.61 Definitions and Basic Properties
  2. 10.62 Graphs
  3. 10.63 Recurrence Relations and Derivatives
  4. 10.64 Integral Representations
  5. 10.65 Power Series
  6. 10.66 Expansions in Series of Bessel Functions
  7. 10.67 Asymptotic Expansions for Large Argument
  8. 10.68 Modulus and Phase Functions
  9. 10.69 Uniform Asymptotic Expansions for Large Order
  10. 10.70 Zeros
  11. 10.71 Integrals
• Applications
  1. 10.72 Mathematical Applications
  2. 10.73 Physical Applications
• Computation
  1. 10.74 Methods of Computation
  2. 10.75 Tables
  3. 10.76 Approximations
  4. 10.77 Software

9.20 Software10.1 Special Notation

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