expansions in modified spherical Bessel functions

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1: 6.10 Other Series Expansions

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§6.10(ii) Expansions in Series of Spherical Bessel Functions

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2: 8.7 Series Expansions

§8.7 Series Expansions

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3: 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. …

4: 10.1 Special Notation

… ►The main functions treated in this chapter are the Bessel functions $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$; Hankel functions $H_{\nu }^{(1)}\left(z\right)$, $H_{\nu }^{(2)}\left(z\right)$; modified Bessel functions $I_{\nu }\left(z\right)$, $K_{\nu }\left(z\right)$; spherical Bessel functions ${𝗃}_n\left(z\right)$, ${𝗒}_n\left(z\right)$, ${𝗁}_n^{(1)}\left(z\right)$, ${𝗁}_n^{(2)}\left(z\right)$; modified spherical Bessel functions ${𝗂}_n^{(1)}\left(z\right)$, ${𝗂}_n^{(2)}\left(z\right)$, ${𝗄}_n\left(z\right)$; Kelvin functions ${\mathrm{ber}}_{\nu }\left(x\right)$, ${\mathrm{bei}}_{\nu }\left(x\right)$, ${\ker }_{\nu }\left(x\right)$, ${\mathrm{kei}}_{\nu }\left(x\right)$. For the spherical Bessel functions and modified spherical Bessel functions the order $n$ is a nonnegative integer. … ►Abramowitz and Stegun (1964): $j_n(z)$, $y_n(z)$, $h_n^{(1)}(z)$, $h_n^{(2)}(z)$, for ${𝗃}_n\left(z\right)$, ${𝗒}_n\left(z\right)$, ${𝗁}_n^{(1)}\left(z\right)$, ${𝗁}_n^{(2)}\left(z\right)$, respectively, when $n\ge 0$. … ►Whittaker and Watson (1927): $K_{\nu }\left(z\right)$ for $\cos \left(\nu \pi \right)K_{\nu }\left(z\right)$. ►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).

5: 7.6 Series Expansions

§7.6 Series Expansions

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§7.6(i) Power Series

… ►The series in this subsection and in §7.6(ii) converge for all finite values of $|z|$. ►

§7.6(ii) Expansions in Series of Spherical Bessel Functions

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7.6.8 $$\mathrm{erf}z=\frac{2z}{\sqrt{\pi }}\sum _{n=0}^{\mathrm{\infty }}{(-1)}^n\left({𝗂}_{2n}^{(1)}\left(z^2\right)-{𝗂}_{2n+1}^{(1)}\left(z^2\right)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.1.7 (in different form)

Referenced by:

§7.6(ii)

Permalink:

http://dlmf.nist.gov/7.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §7.6(ii), §7.6 and Ch.7

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6: 18.15 Asymptotic Approximations

… ►These expansions are in terms of Whittaker functions (§13.14). …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 Bessel Functions

… ►For an error bound for the first term in the Airy-function expansions see Olver (1997b, p. 403). …

7: 33.9 Expansions in Series of Bessel Functions

§33.9 Expansions in Series of Bessel Functions

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§33.9(i) Spherical Bessel Functions

… ►where the function $𝗃$ is as in §10.47(ii), $a_{-1}=0$, $a_0=(2\mathrm{\ell }+1)!!C_{\mathrm{\ell }}\left(\eta \right)$, and … ►

§33.9(ii) Bessel Functions and Modified Bessel Functions

►In this subsection the functions $J$, $I$, and $K$ are as in §§10.2(ii) and 10.25(ii). …

8: 14.15 Uniform Asymptotic Approximations

… ►Here $I$ and $K$ are the modified Bessel functions (§10.25(ii)). … ►For asymptotic expansions and explicit error bounds, see Dunster (2003b). … ►For the Bessel functions $J$ and $Y$ see §10.2(ii), and for the $env$ functions associated with $J$ and $Y$ see §2.8(iv). … ►For convergent series expansions see Dunster (2004). … ►See also Olver (1997b, pp. 311–313) and §18.15(iii) for a generalized asymptotic expansion in terms of elementary functions for Legendre polynomials $P_n\left(\cos \theta \right)$ as $n\to \mathrm{\infty }$ with $\theta$ fixed. …

9: Bibliography S

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A code to evaluate modified Bessel functions based on the continued fraction method. Comput. Phys. Comm. 105 (2-3), pp. 263–272.

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External Links:

ISSN 0010-4655, Document, Code, MathReview, ZentralBlatt

Referenced by:

7th item.

Encodings:

BibTeX

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Bounds for ratios of modified Bessel functions and associated Turán-type inequalities. J. Math. Anal. Appl. 374 (2), pp. 516–528.

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External Links:

ISSN 0022-247X, Document, FullText, MathReview (Beyza Caliskan Aslan), ZentralBlatt

Referenced by:

§10.37, §10.37.

Encodings:

BibTeX

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On the expansion of the parabolic cylinder function in a series of the product of two parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 3, pp. 226–230.

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External Links:

ZentralBlatt

Referenced by:

§12.13(i).

Encodings:

BibTeX

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Uniform asymptotic expansions of modified Mathieu functions. J. Reine Angew. Math. 247, pp. 1–17.

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External Links:

MathReview (J. Meixner), ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

BibTeX

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On the calculation of complex zeros of the modified Bessel function of the second kind. Dokl. Akad. Nauk SSSR 280 (2), pp. 296–299.

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Notes:

English translation in Soviet Math. Dokl. 31(1), pp. 78–81

External Links:

ISSN 0002-3264, MathReview (I. F. Kushnirchuk), ZentralBlatt

Referenced by:

§10.74(vi).

Encodings:

BibTeX

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10: Bibliography B

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Algorithms for evaluating spherical Bessel functions in the complex domain. Zh. Vychisl. Mat. i Mat. Fiz. 28 (12), pp. 1779–1788, 1918.

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Notes:

English translation in U.S.S.R. Comput. Math. and Math. Phys. 28(1988), no. 6, 122–128

External Links:

ISSN 0044-4669, Document, MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

§10.77(iv).

Encodings:

BibTeX

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Asymptotic expansions of the modified Bessel function of the third kind of imaginary order. SIAM J. Appl. Math. 15, pp. 1315–1323.

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External Links:

ISSN 0036-1399, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§10.45.

Encodings:

BibTeX

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The Calculation of Spherical Bessel Functions and Coulomb Functions. In Computational Atomic Physics: Electron and Positron Collisions with Atoms and Ions, K. Bartschat and J. Hinze (Eds.), pp. 181–202.

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Notes:

Includes program disk. Double-precision Fortran code for positive energies. Maximum accuracy: 15D.

External Links:

ISBN 3-540-60179-1, FullText and Code

Referenced by:

3rd item, §33.8.

Encodings:

BibTeX

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Associated Legendre polynomials, ordinary and modified spherical harmonics. Comput. Phys. Comm. 5 (5), pp. 390–394.

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Notes:

Single-precision Fortran, maximum accuracy 16S

External Links:

Document, Code

Referenced by:

2nd item.

Encodings:

BibTeX

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Algorithm 484: Evaluation of the modified Bessel functions K0(Z) and K1(Z) for complex arguments. Comm. ACM 17 (9), pp. 524–526.

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Notes:

Double-precision Fortran code, maximum accuracy 13S.

External Links:

ISSN 0001-0782, Document, Code

Referenced by:

1st item.

Encodings:

BibTeX

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