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A treatise on the theory of Bessel functions. (English) Zbl 0063.08184

Cambridge England: Cambridge University Press; New York: The Macmillan Company; vi, 804 p. (1944).
See the review of the first ed. (1922; JFM 48.0412.02; Zbl 0063.08184).

MSC:

33-02 Research exposition (monographs, survey articles) pertaining to special functions
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
65A05 Tables in numerical analysis
65D20 Computation of special functions and constants, construction of tables

Digital Library of Mathematical Functions:

§10.10 Continued Fractions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
§10.12 Generating Function and Associated Series ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
§10.13 Other Differential Equations ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
§10.14 Inequalities; Monotonicity ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
§10.14 Inequalities; Monotonicity ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
§10.15 Derivatives with Respect to Order ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
Generalized Hypergeometric Functions ‣ §10.16 Relations to Other Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
§10.17(iii) Error Bounds for Real Argument and Order ‣ §10.17 Asymptotic Expansions for Large Argument ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
§10.18(iii) Asymptotic Expansions for Large Argument ‣ §10.18 Modulus and Phase Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
§10.19(ii) Debye’s Expansions ‣ §10.19 Asymptotic Expansions for Large Order ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
§10.19(iii) Transition Region ‣ §10.19 Asymptotic Expansions for Large Order ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
§10.1 Special Notation ‣ Notation ‣ Chapter 10 Bessel Functions
§10.21(iii) Infinite Products ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
§10.21(ii) Analytic Properties ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
§10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
§10.21(iv) Monotonicity Properties ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
§10.21(vi) McMahon’s Asymptotic Expansions for Large Zeros ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
§10.21(xiii) Rayleigh Function ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
(10.22.72) ‣ Triple Products ‣ §10.22(iv) Integrals over the Interval ( 0 , ∞ ) ‣ §10.22 Integrals ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
§10.22(ii) Integrals over Finite Intervals ‣ §10.22 Integrals ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
§10.22(i) Indefinite Integrals ‣ §10.22 Integrals ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
§10.22(iv) Integrals over the Interval ( 0 , ∞ ) ‣ §10.22 Integrals ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
§10.22(vi) Compendia ‣ §10.22 Integrals ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
§10.23(iii) Series Expansions of Arbitrary Functions ‣ §10.23 Sums ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
§10.23(ii) Addition Theorems ‣ §10.23 Sums ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
§10.23(i) Multiplication Theorem ‣ §10.23 Sums ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
Neumann’s Expansion ‣ §10.23(iii) Series Expansions of Arbitrary Functions ‣ §10.23 Sums ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
Fourier–Bessel Expansion ‣ §10.23(iii) Series Expansions of Arbitrary Functions ‣ §10.23 Sums ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
Other Series Expansions ‣ §10.23(iii) Series Expansions of Arbitrary Functions ‣ §10.23 Sums ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
§10.27 Connection Formulas ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions
§10.29(ii) Derivatives ‣ §10.29 Recurrence Relations and Derivatives ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions
§10.29(i) Recurrence Relations ‣ §10.29 Recurrence Relations and Derivatives ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions
§10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
§10.31 Power Series ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions
§10.32(iii) Products ‣ §10.32 Integral Representations ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions
§10.32(ii) Contour Integrals ‣ §10.32 Integral Representations ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions
§10.32(i) Integrals along the Real Line ‣ §10.32 Integral Representations ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions
§10.32(iv) Compendia ‣ §10.32 Integral Representations ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions
§10.34 Analytic Continuation ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions
§10.40(ii) Error Bounds for Real Argument and Order ‣ §10.40 Asymptotic Expansions for Large Argument ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions
§10.40(i) Hankel’s Expansions ‣ §10.40 Asymptotic Expansions for Large Argument ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions
§10.42 Zeros ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions
§10.42 Zeros ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions
§10.43(iv) Integrals over the Interval ( 0 , ∞ ) ‣ §10.43 Integrals ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions
§10.43(vi) Compendia ‣ §10.43 Integrals ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions
Graf’s and Gegenbauer’s Addition Theorems ‣ §10.44(ii) Addition Theorems ‣ §10.44 Sums ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions
§10.4 Connection Formulas ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
§10.54 Integral Representations ‣ Spherical Bessel Functions ‣ Chapter 10 Bessel Functions
§10.60(iii) Other Series ‣ §10.60 Sums ‣ Spherical Bessel Functions ‣ Chapter 10 Bessel Functions
§10.60(iv) Compendia ‣ §10.60 Sums ‣ Spherical Bessel Functions ‣ Chapter 10 Bessel Functions
§10.6(ii) Derivatives ‣ §10.6 Recurrence Relations and Derivatives ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
§10.6(i) Recurrence Relations ‣ §10.6 Recurrence Relations and Derivatives ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
§10.8 Power Series ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
(10.9.26) ‣ §10.9(iii) Products ‣ §10.9 Integral Representations ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
§10.9(iii) Products ‣ §10.9 Integral Representations ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
§10.9(ii) Contour Integrals ‣ §10.9 Integral Representations ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
§10.9(i) Integrals along the Real Line ‣ §10.9 Integral Representations ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
§10.9(iv) Compendia ‣ §10.9 Integral Representations ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions
Chapter 10 Bessel Functions
§11.10(iii) Maclaurin Series ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions
§11.10(ii) Differential Equations ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions
§11.10(i) Definitions ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions
§11.10(ix) Recurrence Relations and Derivatives ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions
§11.10(vi) Relations to Other Functions ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions
§11.10(v) Interrelations ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions
(11.11.10) ‣ §11.11(iii) Large ν , Fixed / z ν ‣ §11.11 Asymptotic Expansions of Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions
(11.11.18) ‣ §11.11(iii) Large ν , Fixed / z ν ‣ §11.11 Asymptotic Expansions of Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions
(11.11.19) ‣ §11.11(iii) Large ν , Fixed / z ν ‣ §11.11 Asymptotic Expansions of Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions
(11.11.8) ‣ §11.11(iii) Large ν , Fixed / z ν ‣ §11.11 Asymptotic Expansions of Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions
§11.11(iii) Large ν , Fixed / z ν ‣ §11.11 Asymptotic Expansions of Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions
§11.11(i) Large | z | , Fixed ν ‣ §11.11 Asymptotic Expansions of Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions
§11.2(ii) Differential Equations ‣ §11.2 Definitions ‣ Struve and Modified Struve Functions ‣ Chapter 11 Struve and Related Functions
§11.2(i) Power-Series Expansions ‣ §11.2 Definitions ‣ Struve and Modified Struve Functions ‣ Chapter 11 Struve and Related Functions
§1.13(v) Products of Solutions ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods
§11.4(ii) Inequalities ‣ §11.4 Basic Properties ‣ Struve and Modified Struve Functions ‣ Chapter 11 Struve and Related Functions
§11.4(v) Recurrence Relations and Derivatives ‣ §11.4 Basic Properties ‣ Struve and Modified Struve Functions ‣ Chapter 11 Struve and Related Functions
§11.5(iii) Compendia ‣ §11.5 Integral Representations ‣ Struve and Modified Struve Functions ‣ Chapter 11 Struve and Related Functions
§11.5(i) Integrals Along the Real Line ‣ §11.5 Integral Representations ‣ Struve and Modified Struve Functions ‣ Chapter 11 Struve and Related Functions
§11.6(iii) Large | ν | , Fixed / z ν ‣ §11.6 Asymptotic Expansions ‣ Struve and Modified Struve Functions ‣ Chapter 11 Struve and Related Functions
§11.6(iii) Large | ν | , Fixed / z ν ‣ §11.6 Asymptotic Expansions ‣ Struve and Modified Struve Functions ‣ Chapter 11 Struve and Related Functions
§11.6(iii) Large | ν | , Fixed / z ν ‣ §11.6 Asymptotic Expansions ‣ Struve and Modified Struve Functions ‣ Chapter 11 Struve and Related Functions
§11.6(i) Large | z | , Fixed ν ‣ §11.6 Asymptotic Expansions ‣ Struve and Modified Struve Functions ‣ Chapter 11 Struve and Related Functions
§11.7(ii) Definite Integrals ‣ §11.7 Integrals and Sums ‣ Struve and Modified Struve Functions ‣ Chapter 11 Struve and Related Functions
§11.9(iii) Asymptotic Expansion ‣ §11.9 Lommel Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions
§11.9(iv) References ‣ §11.9 Lommel Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions
Reflection Formulas ‣ §11.9(i) Definitions ‣ §11.9 Lommel Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions
Chapter 11 Struve and Related Functions
§18.17(v) Fourier Transforms ‣ §18.17 Integrals ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials
Example ‣ §2.5(i) Introduction ‣ §2.5 Mellin Transform Methods ‣ Areas ‣ Chapter 2 Asymptotic Approximations
Erratum (V1.0.21) for Equation (10.22.72) ‣ Errata ‣ Version 1.0.21 (December 15, 2018) ‣ Errata