expansions in spherical Bessel functions

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21: Bibliography B

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L. V. Babushkina, M. K. Kerimov, and A. I. Nikitin (1988b) 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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C. B. Balogh (1967) 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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A. R. Barnett (1996) 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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W. G. C. Boyd (1987) Asymptotic expansions for the coefficient functions that arise in turning-point problems. Proc. Roy. Soc. London Ser. A 410, pp. 35–60.

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External Links:: ISSN 0080-4630, FullText, MathReview (P. F. Hsieh), ZentralBlatt
Referenced by:: §2.8(iii).
Encodings:: BibTeX

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W. G. C. Boyd (1990a) Asymptotic Expansions for the Coefficient Functions Associated with Linear Second-order Differential Equations: The Simple Pole Case. In Asymptotic and Computational Analysis (Winnipeg, MB, 1989), R. Wong (Ed.), Lecture Notes in Pure and Applied Mathematics, Vol. 124, pp. 53–73.

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External Links:: MathReview (John S. Lew), ZentralBlatt
Referenced by:: §2.8(iv).
Encodings:: BibTeX

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22: Errata

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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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Section 14.30

In regard to the definition of the spherical harmonics $Y_{{l},{m}}$ , the domain of the integer $m$ originally written as $0\leq m\leq l$ has been replaced with the more general $|m|\leq l$ . Because of this change, in the sentence just below (14.30.2), “tesseral for $m<l$ and sectorial for $m=l$ ” has been replaced with “tesseral for $|m|<l$ and sectorial for $|m|=l$ ”. Furthermore, in (14.30.4), $m$ has been replaced with $|m|$ .

Reported by Ching-Li Chai on 2019-10-05

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23: Bibliography M

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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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N. W. McLachlan (1961) Bessel Functions for Engineers. 2nd edition, Clarendon Press, Oxford.

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Notes:: The first edition was published in 1934 by Clarendon Press. The second edition was published originally in 1955.
External Links:: ZentralBlatt
Referenced by:: §10.73(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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H. J. W. Müller (1966b) Asymptotic expansions of ellipsoidal wave functions in terms of Hermite functions. Math. Nachr. 32, pp. 49–62.

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External Links:: Document, MathReview (M. J. O. Strutt), ZentralBlatt
Referenced by:: §29.11, §29.7(ii).
Encodings:: BibTeX

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24: Bibliography

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M. Abramowitz (1949) Asymptotic expansions of spheroidal wave functions. J. Math. Phys. Mass. Inst. Tech. 28, pp. 195–199.

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External Links:: MathReview (J. G. van der Corput), ZentralBlatt
Referenced by:: §30.9(iii).
Encodings:: BibTeX

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M. Abramowitz (1954) Regular and irregular Coulomb wave functions expressed in terms of Bessel-Clifford functions. J. Math. Physics 33, pp. 111–116.

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External Links:: MathReview (E. Weber), ZentralBlatt
Referenced by:: §33.9(ii).
Encodings:: BibTeX

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J. C. Adams and P. N. Swarztrauber (1997) SPHEREPACK 2.0: A Model Development Facility. NCAR Technical Note Technical Report TN-436-STR, National Center for Atmospheric Research.

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Notes:: SPHEREPACK 2.0 is a collection of Fortran programs that facilitates computer modeling of geophysical processes. Accurate solutions are obtained via the spectral method that uses both scalar and vector spherical harmonic transforms. The package also contains utility programs for computing the associated Legendre functions. SPHEREPACK 3.1 was released in August 2003.
External Links:: FullText, SPHEREPACK 3.1
Referenced by:: 1st item.
Encodings:: BibTeX

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Z. Altaç (1996) Integrals involving Bickley and Bessel functions in radiative transfer, and generalized exponential integral functions. J. Heat Transfer 118 (3), pp. 789–792.

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External Links:: Document
Referenced by:: §10.73(iv), §8.24(iii).
Encodings:: BibTeX

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R. W. B. Ardill and K. J. M. Moriarty (1978) Spherical Bessel functions

j_{n}

and

y_{n}

of integer order and real argument. Comput. Phys. Comm. 14 (3-4), pp. 261–265.

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Notes:: Single-precision Fortran, maximum accuracy 10S.
External Links:: ISSN 0010-4655, Document, Code
Referenced by:: 1st item.
Encodings:: BibTeX

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25: Bibliography L

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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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W. J. Lentz (1976) Generating Bessel functions in Mie scattering calculations using continued fractions. Applied Optics 15 (3), pp. 668–671.

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Referenced by:: §3.10(iii), §3.10(iii), Erratum (V1.0.24) for Paragraph Steed’s Algorithm (in §3.10(iii)).
Encodings:: BibTeX

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J. L. López and N. M. Temme (2010a) Asymptotics and numerics of polynomials used in Tricomi and Buchholz expansions of Kummer functions. Numer. Math. 116 (2), pp. 269–289.

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External Links:: ISSN 0029-599X, Document, FullText, MathReview (Richard B. Paris), ZentralBlatt
Referenced by:: §13.11, §13.11.
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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26: 18.17 Integrals

… ►and three formulas similar to (18.17.9)–(18.17.11) by symmetry; compare the second row in Table 18.6.1. … ►In particular, in case of exponential Fourier transforms, we may assume

y\in\mathbb{R}

. … ►For the Bessel function

J_{\nu}

see §10.2(ii). … ►In (18.17.21_1) the branch choice of

\sqrt{2c-1}

for

0<c<\frac{1}{2}

is unimportant because on the right-hand side only even powers of

\sqrt{2c-1}

occur after expansion of the Hermite polynomial by (18.5.13). … ►Many of the Fourier transforms given in §18.17(v) have analytic continuations to Laplace transforms. …

27: 18.18 Sums

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Legendre

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Laguerre

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Hermite

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Ultraspherical

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Hermite

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28: Bibliography T

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J. D. Talman (1983) LSFBTR: A subroutine for calculating spherical Bessel transforms. Comput. Phys. Comm. 30 (1), pp. 93–99.

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Notes:: Single-precision Fortran.
External Links:: ISSN 0010-4655, Document, Code
Referenced by:: 9th item.
Encodings:: BibTeX

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N. M. Temme (1979b) The asymptotic expansion of the incomplete gamma functions. SIAM J. Math. Anal. 10 (4), pp. 757–766.

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External Links:: ISSN 0036-1410, Document, MathReview (E. Riekstiņš), ZentralBlatt
Referenced by:: §8.12.
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 (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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I. J. Thompson and A. R. Barnett (1986) Coulomb and Bessel functions of complex arguments and order. J. Comput. Phys. 64 (2), pp. 490–509.

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External Links:: 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)).
Encodings:: BibTeX

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29: 18.34 Bessel Polynomials

… ►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)

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). …

30: Bibliography C

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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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T. M. Cherry (1948) Expansions in terms of parabolic cylinder functions. Proc. Edinburgh Math. Soc. (2) 8, pp. 50–65.

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External Links:: MathReview (I. I. Hirschman, Jr.), ZentralBlatt
Referenced by:: §12.16.
Encodings:: BibTeX

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J. A. Cochran (1966a) The analyticity of cross-product Bessel function zeros. Proc. Cambridge Philos. Soc. 62, pp. 215–226.

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External Links:: MathReview (R. G. Langebartel), ZentralBlatt
Referenced by:: §10.21(x).
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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E. T. Copson (1965) Asymptotic Expansions. Cambridge Tracts in Mathematics and Mathematical Physics, Cambridge University Press, New York.

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External Links:: MathReview (A. E. Danese), ZentralBlatt
Referenced by:: Ch.2.
Encodings:: BibTeX

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