Watson, G. N. 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). Cited in 5 ReviewsCited in 1383 Documents 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 Citations:JFM 48.0412.02; Zbl 0063.08184 × Cite Format Result Cite Review PDF 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 In Triple Products ‣ §10.22(iv) Integrals over the Interval (0,∞) ‣ §10.22 Integrals ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions In §10.22(v) Hankel Transform ‣ §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 In §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 In §11.11(iii) Large 𝜈, Fixed 𝑧/𝜈 ‣ §11.11 Asymptotic Expansions of Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions In §11.11(iii) Large 𝜈, Fixed 𝑧/𝜈 ‣ §11.11 Asymptotic Expansions of Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions In §11.11(iii) Large 𝜈, Fixed 𝑧/𝜈 ‣ §11.11 Asymptotic Expansions of Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions In §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 ‣ Topics of Discussion ‣ 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 In Errata ‣ Version 1.0.21 (December 15, 2018) ‣ Errata