expansions in series of spherical Bessel functions

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11: Bibliography S

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FGH, a code for the calculation of Coulomb radial wave functions from series expansions. Comput. Phys. Comm. 146 (2), pp. 250–253.

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Notes:

Single-precision Fortran code.

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Document, Code, ZentralBlatt

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10th item.

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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).

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On integral representation of Weber’s parabolic cylinder function and its expansion into an infinite series. J. Indian Math. Soc. (N. S.) 4, pp. 34–38.

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External Links:

MathReview (M. C. Gray), ZentralBlatt

Referenced by:

§12.13(ii).

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Numerical evaluation of spherical Bessel transforms via fast Fourier transforms. J. Comput. Phys. 100 (2), pp. 294–296.

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ISSN 0021-9991, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vii).

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The expansion of Lamé functions into series of associated Legendre functions of the second kind. Proc. Cambridge Philos. Soc. 62, pp. 441–452.

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Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.17(i).

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12: Bibliography H

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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).

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

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ISSN 0168-9274, Document, MathReview, ZentralBlatt

Referenced by:

§10.60(iii), §6.10(ii), §6.15.

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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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ISSN 1364-5021, Document, MathReview (A. D. Wood), ZentralBlatt

Referenced by:

§10.20(ii).

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

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ISSN 0022-2488, Document, MathReview, ZentralBlatt

Referenced by:

§33.10(ii), §33.10(iii), §33.14(iv), §33.9(ii), §33.9(ii).

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Asymptotic expansions of Mathieu functions in wave mechanics. J. Comput. Phys. 21 (3), pp. 319–325.

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Document, MathReview (John P. Coleman)

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5th item.

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13: Bibliography D

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Complex zeros of linear combinations of spherical Bessel functions and their derivatives. SIAM J. Math. Anal. 4 (1), pp. 128–133.

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ISSN 1095-7154, Document, MathReview (C. J. Bouwkamp), ZentralBlatt

Referenced by:

§10.58.

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Chebyshev series for the spherical Bessel function $j_l(r)$ . Comput. Phys. Comm. 18 (1), pp. 73–86.

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Document

Referenced by:

§10.76(iii).

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Convergent Liouville-Green expansions for second-order linear differential equations, with an application to Bessel functions. Proc. Roy. Soc. London Ser. A 440, pp. 37–54.

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ISSN 0962-8444, FullText, MathReview (W. Balser), ZentralBlatt

Referenced by:

§10.41(iii), §2.8(i).

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Convergent expansions for solutions of linear ordinary differential equations having a simple turning point, with an application to Bessel functions. Stud. Appl. Math. 107 (3), pp. 293–323.

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ISSN 0022-2526, Document, MathReview, ZentralBlatt

Referenced by:

§10.20(ii), §2.8(i).

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Uniform asymptotic expansions for the reverse generalized Bessel polynomials, and related functions. SIAM J. Math. Anal. 32 (5), pp. 987–1013.

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ISSN 0036-1410, Document, MathReview (Hasan Taşeli), ZentralBlatt

Referenced by:

§10.49(ii), §18.34(iii).

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14: Bibliography

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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).

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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).

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Modular Functions and Dirichlet Series in Number Theory. 2nd edition, Graduate Texts in Mathematics, Vol. 41, Springer-Verlag, New York.

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ISBN 0-387-97127-0, MathReview, ZentralBlatt

Referenced by:

§17.2(i), §23.10(iv), §23.15(i), §23.17(ii), §23.18, §23.18, §23.19, §23.20(i), §23.20(v), Ch.23, §27.14(iii), §27.14(iv), §27.14(vi), §27.7, Ch.27.

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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.

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ISSN 0010-4655, Document, Code

Referenced by:

1st item.

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Orthogonal Polynomials and Special Functions. CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 21, Society for Industrial and Applied Mathematics, Philadelphia, PA.

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ISBN 0-89871-018-9, MathReview (G. Gasper), ZentralBlatt

Referenced by:

§18.10(ii), §18.18(vii), Profile Richard A. Askey.

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15: Errata

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18.34.2

$y_n(x)=y_n(x;2)=2{\pi }^{-1}{x}^{-1}{\mathrm{e}}^{1/x}{𝗄}_n\left(x^{-1}\right),$

${\theta }_n(x)=x^n{y}_n(x^{-1})=2{\pi }^{-1}{x}^{n+1}{\mathrm{e}}^x{𝗄}_n\left(x\right)$

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

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The following additions were made in Chapter 1:

• 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).

• 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).

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

• 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).

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

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

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

• 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.

• 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.

• 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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§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 $\mathrm{Wp}\left(x\right)=W_0\left(x\right)$ and $\mathrm{Wm}\left(x\right)=W_{\pm 1}\left(x\mp 0\mathrm{i}\right)$.

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 $\frac{d^{n}W}{{dz}^n}$, 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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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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It is now mentioned that (25.2.5), defines the Stieltjes constants ${\gamma }_n$. Consequently, ${\gamma }_n$ in (25.2.4), (25.6.12) are now identified as the Stieltjes constants.

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16: 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).

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

Referenced by:

§10.45.

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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.

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

Referenced by:

3rd item, §33.8.

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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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ISSN 0080-4630, FullText, MathReview (P. F. Hsieh), ZentralBlatt

Referenced by:

§2.8(iii).

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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).

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17: Bibliography M

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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.

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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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ISSN 0022-2488, Document, MathReview (S. K. Chatterjea), ZentralBlatt

Referenced by:

§1.17(ii), §10.59.

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Analytic expressions for integrals of products of spherical Bessel functions. J. Phys. A 24 (7), pp. 1435–1453.

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ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§10.59.

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On the convergence of the Chebyshev series for functions possessing a singularity in the range of representation. SIAM J. Numer. Anal. 3 (3), pp. 390–409.

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ISSN 1095-7170, Document, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§3.11(ii).

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A $q$-analog of hypergeometric series well-poised in $\mathrm{𝑆𝑈}(n)$ and invariant $G$-functions. Adv. in Math. 58 (1), pp. 1–60.

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ISSN 0001-8708, Document, MathReview (Robert A. Gustafson), ZentralBlatt

Referenced by:

§17.15.

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18: 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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19: Bibliography T

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

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Notes:

Single-precision Fortran.

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ISSN 0010-4655, Document, Code

Referenced by:

9th item.

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

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ISSN 0036-1410, Document, MathReview (E. Riekstiņš), ZentralBlatt

Referenced by:

§8.12.

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

Referenced by:

§13.8(iii), §13.8(iii).

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Computational aspects of incomplete gamma functions with large complex parameters. In Approximation and Computation. A Festschrift in Honor of Walter Gautschi, R. V. M. Zahar (Ed.), International Series of Numerical Mathematics, Vol. 119, pp. 551–562.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§8.11(iii), §8.20(ii), §8.25(i), §8.25(iii), 2nd item, 3rd item.

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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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20: Bibliography C

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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.

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Power series for inverse Jacobian elliptic functions. Math. Comp. 77 (263), pp. 1615–1621.

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ISSN 0025-5718, Document, MathReview (Shaun Cooper), ZentralBlatt

Referenced by:

§19.25(v), §22.15(ii).

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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.

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Chebyshev expansions for the Bessel function $J_n(z)$ in the complex plane. Math. Comp. 40 (161), pp. 343–366.

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ISSN 0025-5718, FullText, MathReview (C. W. Clenshaw), ZentralBlatt

Referenced by:

§10.76(ii).

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A sequence of series for the Lambert $W$ function. In Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation (Kihei, HI), pp. 197–204.

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External Links:

Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

(4.13.4_1), (4.13.4_2), (4.13.5_2), §4.13.

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