DLMF: Untitled Document

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P. I. Hadži (1976a) Expansions for the probability function in series of Čebyšev polynomials and Bessel functions. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).

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External Links:: MathReview (V. L. Makarov)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

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F. E. Harris (2000) Spherical Bessel expansions of sine, cosine, and exponential integrals. Appl. Numer. Math. 34 (1), pp. 95–98.

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External Links:: ISSN 0168-9274, Document, MathReview, ZentralBlatt
Referenced by:: §10.60(iii), §6.10(ii), §6.15.
Encodings:: BibTeX

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C. J. Howls and A. B. Olde Daalhuis (1999) On the resurgence properties of the uniform asymptotic expansion of Bessel functions of large order. Proc. Roy. Soc. London Ser. A 455, pp. 3917–3930.

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External Links:: ISSN 1364-5021, Document, MathReview (A. D. Wood), ZentralBlatt
Referenced by:: §10.20(ii).
Encodings:: BibTeX

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J. Humblet (1985) Bessel function expansions of Coulomb wave functions. J. Math. Phys. 26 (4), pp. 656–659.

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External Links:: ISSN 0022-2488, Document, MathReview, ZentralBlatt
Referenced by:: §33.10(ii), §33.10(iii), §33.14(iv), §33.9(ii), §33.9(ii).
Encodings:: BibTeX

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G. Hunter and M. Kuriyan (1976) Asymptotic expansions of Mathieu functions in wave mechanics. J. Comput. Phys. 21 (3), pp. 319–325.

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External Links:: Document, MathReview (John P. Coleman)
Referenced by:: 5th item.
Encodings:: BibTeX

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§10.75(v) Modified Bessel Functions and their Derivatives

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§10.75(vi) Zeros of Modified Bessel Functions and their Derivatives

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§10.75(viii) Modified Bessel Functions of Imaginary or Complex Order

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§10.75(ix) Spherical Bessel Functions, Modified Spherical Bessel Functions, and their Derivatives

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§10.75(x) Zeros and Associated Values of Derivatives of Spherical Bessel Functions

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A. Laforgia (1991) Bounds for modified Bessel functions. J. Comput. Appl. Math. 34 (3), pp. 263–267.

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External Links:: ISSN 0377-0427, Document, MathReview (N. Hayek Calil), ZentralBlatt
Referenced by:: §10.37.
Encodings:: BibTeX

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D. R. Lehman, W. C. Parke, and L. C. Maximon (1981) Numerical evaluation of integrals containing a spherical Bessel function by product integration. J. Math. Phys. 22 (7), pp. 1399–1413.

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External Links:: ISSN 0022-2488, Document, MathReview (C. W. Clenshaw), ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

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D. Lemoine (1997) Optimal cylindrical and spherical Bessel transforms satisfying bound state boundary conditions. Comput. Phys. Comm. 99 (2-3), pp. 297–306.

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Notes:: Single-precision Fortran, maximum accuracy 8S.
External Links:: ISSN 0010-4655, Document, Code, ZentralBlatt
Referenced by:: 4th item.
Encodings:: BibTeX

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K. V. Leung and S. S. Ghaderpanah (1979) An application of the finite element approximation method to find the complex zeros of the modified Bessel function

K_{n}(z)

. Math. Comp. 33 (148), pp. 1299–1306.

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External Links:: ISSN 0025-5718, FullText, MathReview (R. P. Boas, Jr.), ZentralBlatt
Referenced by:: §10.74(vi), 2nd item.
Encodings:: BibTeX

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Y. L. Luke (1959) Expansion of the confluent hypergeometric function in series of Bessel functions. Math. Tables Aids Comput. 13 (68), pp. 261–271.

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External Links:: FullText, MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §13.24(ii).
Encodings:: BibTeX

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A. L. Van Buren and J. E. Boisvert (2007) Accurate calculation of the modified Mathieu functions of integer order. Quart. Appl. Math. 65 (1), pp. 1–23.

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Notes:: For software, see Van Buren (website)
External Links:: ISSN 0033-569X, MathReview (Javier Segura), ZentralBlatt
Referenced by:: §28.36(iii).
Encodings:: BibTeX

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B. Ph. van Milligen and A. López Fraguas (1994) Expansion of vacuum magnetic fields in toroidal harmonics. Comput. Phys. Comm. 81 (1-2), pp. 74–90.

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External Links:: Document
Referenced by:: §14.31(i).
Encodings:: BibTeX

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A. N. Vavreck and W. Thompson (1984) Some novel infinite series of spherical Bessel functions. Quart. Appl. Math. 42 (3), pp. 321–324.

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External Links:: ISSN 0033-569X, MathReview (A. de Castro Brzezicki), ZentralBlatt
Referenced by:: §10.60(iii), §10.60(iii).
Encodings:: BibTeX

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R. Vidūnas and N. M. Temme (2002) Symbolic evaluation of coefficients in Airy-type asymptotic expansions. J. Math. Anal. Appl. 269 (1), pp. 317–331.

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External Links:: ISSN 0022-247X, Document, MathReview, ZentralBlatt
Referenced by:: §2.4(v).
Encodings:: BibTeX

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H. Volkmer (1999) Expansions in products of Heine-Stieltjes polynomials. Constr. Approx. 15 (4), pp. 467–480.

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External Links:: ISSN 0176-4276, Document, MathReview (Luoqing Li), ZentralBlatt
Referenced by:: §31.15(iii).
Encodings:: BibTeX

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N. M. Temme (1975) On the numerical evaluation of the modified Bessel function of the third kind. J. Comput. Phys. 19 (3), pp. 324–337.

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Notes:: Algol program.
External Links:: Document, MathReview (Carl C. Grosjean), ZentralBlatt
Referenced by:: §10.74(i), §10.74(iv), §10.77(iii).
Encodings:: BibTeX

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N. M. Temme (1990b) 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).
Encodings:: BibTeX

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N. M. Temme (1994c) 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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External Links:: ISSN 1073-2772, FullText, MathReview (Anatoly Kilbas), ZentralBlatt
Referenced by:: §10.74(viii).
Encodings:: BibTeX

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N. M. Temme (1978) The numerical computation of special functions by use of quadrature rules for saddle point integrals. II. Gamma functions, modified Bessel functions and parabolic cylinder functions. Report TW 183/78 Mathematisch Centrum, Amsterdam, Afdeling Toegepaste Wiskunde.

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Notes:: Algol procedures, maximum accuracy 14S.
External Links:: FullText, ZentralBlatt
Referenced by:: §12.21(ii).
Encodings:: BibTeX

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M. J. Tretter and G. W. Walster (1980) Further comments on the computation of modified Bessel function ratios. Math. Comp. 35 (151), pp. 937–939.

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External Links:: ISSN 0025-5718, FullText, Document, MathReview (James M. Horner), ZentralBlatt
Referenced by:: §10.74(v).
Encodings:: BibTeX

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… ►where

\mathsf{k}_{n}

is a modified spherical Bessel function (10.49.9), and … ►expressed in terms of Romanovski–Bessel polynomials, Laguerre polynomials or Whittaker functions, we have …In this limit the finite system of Jacobi polynomials

P^{(\alpha,\beta)}_{n}\left(x\right)

which is orthogonal on

(1,\infty)

(see §18.3) tends to the finite system of Romanovski–Bessel polynomials which is orthogonal on

(0,\infty)

(see (18.34.5_5)). ►For uniform asymptotic expansions of

y_{n}\left(x;a\right)

as

n\to\infty

in terms of Airy functions (§9.2) see Wong and Zhang (1997) and Dunster (2001c). For uniform asymptotic expansions in terms of Hermite polynomials see López and Temme (1999b). …

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A. J. MacLeod (1993) Chebyshev expansions for modified Struve and related functions. Math. Comp. 60 (202), pp. 735–747.

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External Links:: ISSN 0025-5718, FullText, MathReview, ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

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T. M. MacRobert (1967) Spherical Harmonics. An Elementary Treatise on Harmonic Functions with Applications. 3rd edition, International Series of Monographs in Pure and Applied Mathematics, Vol. 98, Pergamon Press, Oxford.

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External Links:: MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: Ch.14.
Encodings:: BibTeX

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L. C. Maximon (1991) On the evaluation of the integral over the product of two spherical Bessel functions. J. Math. Phys. 32 (3), pp. 642–648.

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External Links:: ISSN 0022-2488, Document, MathReview (S. K. Chatterjea), ZentralBlatt
Referenced by:: §1.17(ii), §10.59.
Encodings:: BibTeX

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Fr. Mechel (1966) Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

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External Links:: ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt
Referenced by:: §10.74(iii).
Encodings:: BibTeX

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R. Mehrem, J. T. Londergan, and M. H. Macfarlane (1991) Analytic expressions for integrals of products of spherical Bessel functions. J. Phys. A 24 (7), pp. 1435–1453.

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External Links:: ISSN 0305-4470, Document, MathReview, ZentralBlatt
Referenced by:: §10.59.
Encodings:: BibTeX

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J. B. Campbell (1980) On Temme’s algorithm for the modified Bessel function of the third kind. ACM Trans. Math. Software 6 (4), pp. 581–586.

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External Links:: ISSN 0098-3500, Document, MathReview, ZentralBlatt
Referenced by:: §10.74(iv).
Encodings:: BibTeX

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B. C. Carlson and G. S. Rushbrooke (1950) On the expansion of a Coulomb potential in spherical harmonics. Proc. Cambridge Philos. Soc. 46, pp. 626–633.

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External Links:: Document, MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §34.12.
Encodings:: BibTeX

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R. Cicchetti and A. Faraone (2004) Incomplete Hankel and modified Bessel functions: A class of special functions for electromagnetics. IEEE Trans. Antennas and Propagation 52 (12), pp. 3373–3389.

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External Links:: ISSN 0018-926X, Document, MathReview (T. Erber), ZentralBlatt
Referenced by:: §10.46.
Encodings:: BibTeX

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W. J. Cody (1983) Algorithm 597: Sequence of modified Bessel functions of the first kind. ACM Trans. Math. Software 9 (2), pp. 242–245.

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Notes:: Double-precision Fortran, maximum accuracy 24S.
External Links:: ISSN 0098-3500, Document, GAMS (module 597; package TOMS), ZentralBlatt
Referenced by:: 5th item.
Encodings:: BibTeX

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J. P. Coleman and A. J. Monaghan (1983) Chebyshev expansions for the Bessel function

J_{n}(z)

in the complex plane. Math. Comp. 40 (161), pp. 343–366.

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External Links:: ISSN 0025-5718, FullText, MathReview (C. W. Clenshaw), ZentralBlatt
Referenced by:: §10.76(ii).
Encodings:: BibTeX

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Legendre

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Laguerre

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Hermite

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Laguerre

… ►For the modified Bessel function

I_{\nu}\left(z\right)

see §10.25(ii). …

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Equation (18.34.2)

18.34.2

y_{n}(x)=y_{n}\left(x;2\right)=2{\pi}^{-1}x^{-1}{\mathrm{e}}^{1/x}\mathsf{k}_{% n}\left(x^{-1}\right),

\theta_{n}(x)=x^{n}y_{n}(x^{-1})=2{\pi}^{-1}x^{n+1}{\mathrm{e}}^{x}\mathsf{k}_% {n}\left(x\right)

This equation was updated to include definitions in terms of the modified spherical Bessel function of the second kind.

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Chapter 1 Additions

The following additions were made in Chapter 1:

Section 1.2

New subsections, 1.2(v) Matrices, Vectors, Scalar Products, and Norms and 1.2(vi) Square Matrices, with Equations (1.2.27)–(1.2.77).
Section 1.3

The title of this section was changed from “Determinants” to “Determinants, Linear Operators, and Spectral Expansions”. An extra paragraph just below (1.3.7). New subsection, 1.3(iv) Matrices as Linear Operators, with Equations (1.3.20), (1.3.21).
Section 1.4

In 1.4(v), three new paragraphs on Stieltjes integrals/measure, with Equations (1.4.23_1)–(1.4.23_3).
Section 1.8

In Subsection 1.8(i), the title of the paragraph “Bessel’s Inequality” was changed to “Parseval’s Formula”. We give the relation between the real and the complex coefficients, and include more general versions of Parseval’s Formula, Equations (1.8.6_1), (1.8.6_2). The title of Subsection 1.8(iv) was changed from “Transformations” to “Poisson’s Summation Formula”, and we added an extra remark just below (1.8.14).
Section 1.10

New subsection, 1.10(xi) Generating Functions, with Equations (1.10.26)–(1.10.29).
Section 1.13

New subsection, 1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms, with Equations (1.13.26)–(1.13.31).
Section 1.14(i)

Another form of Parseval’s formula, (1.14.7_5).
Section 1.16

We include several extra remarks and Equations (1.16.3_5), (1.16.9_5). New subsection, 1.16(ix) References for Section 1.16.
Section 1.17

Two extra paragraphs in Subsection 1.17(ii) Integral Representations, with Equations (1.17.12_1), (1.17.12_2); Subsection 1.17(iv) Mathematical Definitions is almost completely rewritten.
Section 1.18

An entire new section, 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions, including new subsections, 1.18(i)–1.18(x), and several equations, (1.18.1)–(1.18.71).

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Expansion

§4.13 has been enlarged. The Lambert $W$ -function is multi-valued and we use the notation $W_{k}\left(x\right)$ , $k\in\mathbb{Z}$ , for the branches. The original two solutions are identified via $\operatorname{Wp}\left(x\right)=W_{0}\left(x\right)$ and $\operatorname{Wm}\left(x\right)=W_{\pm 1}\left(x\mp 0\mathrm{i}\right)$ .

Other changes are the introduction of the Wright $\omega$ -function and tree $T$ -function in (4.13.1_2) and (4.13.1_3), simplification formulas (4.13.3_1) and (4.13.3_2), explicit representation (4.13.4_1) for $\frac{{\mathrm{d}}^{n}W}{{\mathrm{d}z}^{n}}$ , additional Maclaurin series (4.13.5_1) and (4.13.5_2), an explicit expansion about the branch point at $z=-{\mathrm{e}}^{-1}$ in (4.13.9_1), extending the number of terms in asymptotic expansions (4.13.10) and (4.13.11), and including several integrals and integral representations for Lambert $W$ -functions in the end of the section.

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Chapters 10 Bessel Functions, 18 Orthogonal Polynomials, 34 3j, 6j, 9j Symbols

The Legendre polynomial $P_{n}$ was mistakenly identified as the associated Legendre function $P_{n}$ in §§10.54, 10.59, 10.60, 18.18, 18.41, 34.3 (and was thus also affected by the bug reported below). These symbols now link correctly to their definitions. Reported by Roy Hughes on 2022-05-23

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References

Bibliographic citations and clarifications have been added, removed, or modified in §§5.6(i), 5.10, 7.8, 7.25(iii), and 32.16.

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expansions in modified spherical Bessel functions

11—20 of 21 matching pages

11: Bibliography H

12: 10.75 Tables

§10.75(v) Modified Bessel Functions and their Derivatives

§10.75(vi) Zeros of Modified Bessel Functions and their Derivatives

§10.75(viii) Modified Bessel Functions of Imaginary or Complex Order

§10.75(ix) Spherical Bessel Functions, Modified Spherical Bessel Functions, and their Derivatives

§10.75(x) Zeros and Associated Values of Derivatives of Spherical Bessel Functions

13: Bibliography L

14: Bibliography V

15: Bibliography T

16: 18.34 Bessel Polynomials

17: Bibliography M

18: Bibliography C

19: 18.18 Sums

Legendre

Laguerre

Hermite

Laguerre

20: Errata