in terms of Bessel functions of variable order

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11: 10.17 Asymptotic Expansions for Large Argument

§10.17 Asymptotic Expansions for Large Argument

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§10.17(iii) Error Bounds for Real Argument and Order

… ►If these expansions are terminated when $k=\mathrm{\ell }-1$, then the remainder term is bounded in absolute value by the first neglected term, provided that $\mathrm{\ell }\ge \max (\nu -\frac{1}{2},1)$. … ►For higher re-expansions of the remainder terms see Olde Daalhuis and Olver (1995a) and Olde Daalhuis (1995, 1996).

12: 18.15 Asymptotic Approximations

… ►The latter expansions are in terms of Bessel functions, and are uniform in complex $z$-domains not containing neighborhoods of 1. … ►These expansions are in terms of Bessel functions and modified Bessel functions, respectively. … ►

In Terms of Elementary Functions

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In Terms of Bessel Functions

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In Terms of Airy Functions

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13: 10.41 Asymptotic Expansions for Large Order

§10.41 Asymptotic Expansions for Large Order

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§10.41(ii) Uniform Expansions for Real Variable

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14: 13.9 Zeros

… ►When and $b>0$ let ${\varphi }_r$, $r=1,2,3,\mathrm{\dots }$, be the positive zeros of $M(a,b,x)$ arranged in increasing order of magnitude, and let $j_{b-1,r}$ be the $r$th positive zero of the Bessel function $J_{b-1}\left(x\right)$ (§10.21(i)). … … ► … ► ►For fixed $a$ and $z$ in $ℂ$ the function $M(a,b,z)$ has only a finite number of $b$-zeros. …

15: 10.22 Integrals

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Orthogonality

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Orthogonality

… ►See also §1.17(ii) for an integral representation of the Dirac delta in terms of a product of Bessel functions. … ►Additional infinite integrals over the product of three Bessel functions (including modified Bessel functions) are given in Gervois and Navelet (1984, 1985a, 1985b, 1986a, 1986b). … ►For collections of integrals of the functions $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$, $H_{\nu }^{(1)}\left(z\right)$, and $H_{\nu }^{(2)}\left(z\right)$, including integrals with respect to the order, see Andrews et al. (1999, pp. 216–225), Apelblat (1983, §12), Erdélyi et al. (1953b, §§7.7.1–7.7.7 and 7.14–7.14.2), Erdélyi et al. (1954a, b), Gradshteyn and Ryzhik (2000, §§5.5 and 6.5–6.7), Gröbner and Hofreiter (1950, pp. 196–204), Luke (1962), Magnus et al. (1966, §3.8), Marichev (1983, pp. 191–216), Oberhettinger (1974, §§1.10 and 2.7), Oberhettinger (1990, §§1.13–1.16 and 2.13–2.16), Oberhettinger and Badii (1973, §§1.14 and 2.12), Okui (1974, 1975), Prudnikov et al. (1986b, §§1.8–1.10, 2.12–2.14, 3.2.4–3.2.7, 3.3.2, and 3.4.1), Prudnikov et al. (1992a, §§3.12–3.14), Prudnikov et al. (1992b, §§3.12–3.14), Watson (1944, Chapters 5, 12, 13, and 14), and Wheelon (1968).

16: Bibliography T

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Uniform asymptotic expansions of a class of integrals in terms of modified Bessel functions, with application to confluent hypergeometric functions. SIAM J. Math. Anal. 21 (1), pp. 241–261.

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External Links:

ISSN 0036-1410, Document, MathReview (Pablo Martín), ZentralBlatt

Referenced by:

§13.8(iii), §13.8(iii).

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Steepest descent paths for integrals defining the modified Bessel functions of imaginary order. Methods Appl. Anal. 1 (1), pp. 14–24.

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ISSN 1073-2772, FullText, MathReview (Anatoly Kilbas), ZentralBlatt

Referenced by:

§10.74(viii).

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Coulomb and Bessel functions of complex arguments and order. J. Comput. Phys. 64 (2), pp. 490–509.

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ISSN 0021-9991, Document, MathReview (R. C. Varma), ZentralBlatt

Referenced by:

§10.74(v), §13.32(iii), §3.10(iii), §3.10(iii), §33.13, §33.23(vi), §33.26(iii), Ch.33, Erratum (V1.0.24) for Paragraph Steed’s Algorithm (in §3.10(iii)).

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Modified Bessel functions $I_{\nu }(z)$ and $K_{\nu }(z)$ of real order and complex argument, to selected accuracy. Comput. Phys. Comm. 47 (2-3), pp. 245–257.

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Notes:

For erratum see same journal 159 (2004), no. 3, p. 243. Double-precision Fortran, accuracy 24S, but can be increased.

External Links:

ISSN 0010-4655, Document, Code, MathReview (John P. Coleman)

Referenced by:

3rd item, 4th item.

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Erratum to “COULCC: A continued-fraction algorithm for Coulomb functions of complex order with complex arguments”. Comput. Phys. Comm. 159 (3), pp. 241–242.

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External Links:

Document, Code, ZentralBlatt

Referenced by:

§33.23(vi), I. J. Thompson and A. R. Barnett (1985).

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17: 11.10 Anger–Weber Functions

§11.10 Anger–Weber Functions

… ►The Anger and Weber functions satisfy the inhomogeneous Bessel differential equation … ►

§11.10(vi) Relations to Other Functions

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§11.10(viii) Expansions in Series of Products of Bessel Functions

… ►where the prime on the second summation symbols means that the first term is to be halved. …

18: 18.39 Applications in the Physical Sciences

… ►The functions ${\varphi }_n$ are expressed in terms of Romanovski–Bessel polynomials, or Laguerre polynomials by (18.34.7_1). … ►

a) Spherical Radial Coulomb Wave Functions Expressed in terms of Laguerre OP’s

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d) Radial Coulomb Wave Functions Expressed in Terms of the Associated Coulomb–Laguerre OP’s

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Discretized and Continuum Expansions of Scattering Eigenfunctions in terms of Pollaczek Polynomials: J-matrix Theory

… ►The fact that non-$L^2$ continuum scattering eigenstates may be expressed in terms or (infinite) sums of $L^2$ functions allows a reformulation of scattering theory in atomic physics wherein no non-$L^2$ functions need appear. …

19: Bibliography W

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Tables of Summable Series and Integrals Involving Bessel Functions. Holden-Day, San Francisco, CA.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§10.22(vi), §10.23(iv), §10.43(vi).

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Algorithm 44: Bessel functions computed recursively. Comm. ACM 4 (4), pp. 177–178.

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Notes:

Includes Algol code, maximum accuracy 8S.

External Links:

ISSN 0001-0782, Document

Referenced by:

§10.77(ii).

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On the points of inflection of Bessel functions of positive order. II. Canad. J. Math. 43 (3), pp. 628–651.

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External Links:

ISSN 0008-414X, MathReview (Evangelos Ifantis), ZentralBlatt

Referenced by:

§10.21(iv).

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Estimates for the error term in a uniform asymptotic expansion of the Jacobi polynomials. Anal. Appl. (Singap.) 1 (2), pp. 213–241.

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External Links:

ISSN 0219-5305, Document, MathReview (Eberhard Wagner), ZentralBlatt

Referenced by:

§18.15(i), §18.15(i).

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The generalized Bessel function of order greater than one. Quart. J. Math., Oxford Ser. 11, pp. 36–48.

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External Links:

Document, MathReview (M. C. Gray), ZentralBlatt

Referenced by:

§10.46.

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20: 10.74 Methods of Computation

… ►In the case of the spherical Bessel functions the explicit formulas given in §§10.49(i) and 10.49(ii) are terminating cases of the asymptotic expansions given in §§10.17(i) and 10.40(i) for the Bessel functions and modified Bessel functions. … ►If values of the Bessel functions $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$, or the other functions treated in this chapter, are needed for integer-spaced ranges of values of the order $\nu$, then a simple and powerful procedure is provided by recurrence relations typified by the first of (10.6.1). … ►

Fourier–Bessel Expansion

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§10.74(viii) Functions of Imaginary Order

►For the computation of the functions ${\tilde{I}}_{\nu }\left(x\right)$ and ${\tilde{K}}_{\nu }\left(x\right)$ defined by (10.45.2) see Temme (1994c) and Gil et al. (2002d, 2003a, 2004b).