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11: Bibliography W

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Rational approximations for the modified Bessel function of the second kind. Comput. Phys. Comm. 59 (3), pp. 471–493.

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External Links:

ISSN 0010-4655, Document, MathReview, ZentralBlatt

Referenced by:

§10.76(ii).

Encodings:

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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).

Encodings:

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On a generalization of the functions $\mathrm{ber}$ x, $\mathrm{bei}$ x, $\ker$ x, $\mathrm{kei}$ x. Quart. J. Pure Appl. Math. 42, pp. 316–342.

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External Links:

ZentralBlatt

Referenced by:

§10.61(iii), §10.61(iv), §10.65(i), §10.65(ii), §10.65(iii), §10.68(iii).

Encodings:

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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).

Encodings:

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The asymptotic expansion of the generalized Bessel function. Proc. London Math. Soc. (2) 38, pp. 257–270.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.46.

Encodings:

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12: Bibliography M

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On the zeros of cross-product Bessel functions. J. Math. Mech. 16, pp. 447–452.

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External Links:

MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§10.21(x).

Encodings:

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On the evaluation of Bessel functions. Appl. Comput. Harmon. Anal. 1 (1), pp. 116–135.

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External Links:

ISSN 1063-5203, Document, MathReview, ZentralBlatt

Referenced by:

§10.77(iii).

Encodings:

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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.

Encodings:

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Inequalities for the zeros of Bessel functions. SIAM J. Math. Anal. 8 (1), pp. 166–170.

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ISSN 1095-7154, Document, MathReview (Mary L. Boas), ZentralBlatt

Referenced by:

§10.21(iv).

Encodings:

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Singular integrals whose kernels involve certain Sturm-Liouville functions. I. J. Math. Mech. 19 (10), pp. 855–873.

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External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§9.10(ii).

Encodings:

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13: 10.75 Tables

§10.75 Tables

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§10.75(iv) Integrals of Bessel Functions

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

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Young and Kirk (1964) tabulates ${\mathrm{ber}}_{n}x$, ${\mathrm{bei}}_{n}x$, ${\ker }_{n}x$, ${\mathrm{kei}}_{n}x$, $n=0,1$, $x=0(.1)10$, 15D; ${\mathrm{ber}}_{n}x$, ${\mathrm{bei}}_{n}x$, ${\ker }_{n}x$, ${\mathrm{kei}}_{n}x$, modulus and phase functions $M_n\left(x\right)$, ${\theta }_n\left(x\right)$, $N_n\left(x\right)$, ${\varphi }_n\left(x\right)$, $n=0,1,2$, $x=0(.01)2.5$, 8S, and $n=0(1)10$, $x=0(.1)10$, 7S. Also included are auxiliary functions to facilitate interpolation of the tables for $n=0(1)10$ for small values of $x$. (Concerning the phase functions see §10.68(iv).)

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Abramowitz and Stegun (1964, Chapter 9) tabulates ${\mathrm{ber}}_{n}x$, ${\mathrm{bei}}_{n}x$, ${\ker }_{n}x$, ${\mathrm{kei}}_{n}x$, $n=0,1$, $x=0(.1)5$, 9–10D; $x^n({\ker }_{n}x+({\mathrm{ber}}_{n}x)(\ln x))$, $x^n({\mathrm{kei}}_{n}x+({\mathrm{bei}}_{n}x)(\ln x))$, $n=0,1$, $x=0(.1)1$, 9D; modulus and phase functions $M_n\left(x\right)$, ${\theta }_n\left(x\right)$, $N_n\left(x\right)$, ${\varphi }_n\left(x\right)$, $n=0,1$, $x=0(.2)7$, 6D; $\sqrt{x}{\mathrm{e}}^{-x/\sqrt{2}}{M}_n\left(x\right)$, ${\theta }_n\left(x\right)-(x/\sqrt{2})$, $\sqrt{x}{\mathrm{e}}^{x/\sqrt{2}}{N}_n\left(x\right)$, ${\varphi }_n\left(x\right)+(x/\sqrt{2})$, $n=0,1$, $1/x=0(.01)0.15$, 5D.

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14: Bibliography T

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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:

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On the numerical evaluation of the ordinary Bessel function of the second kind. J. Computational Phys. 21 (3), pp. 343–350.

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Notes:

Algol program

External Links:

ISSN 0021-9991, Document, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§10.77(iii).

Encodings:

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An algorithm with ALGOL 60 program for the computation of the zeros of ordinary Bessel functions and those of their derivatives. J. Comput. Phys. 32 (2), pp. 270–279.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.74(vi), §10.77(x).

Encodings:

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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:

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Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 (1), pp. 151–174.

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External Links:

ISSN 0010-3616, FullText, MathReview (Estelle L. Basor), ZentralBlatt

Referenced by:

§32.14.

Encodings:

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15: 18.12 Generating Functions

§18.12 Generating Functions

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Jacobi

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Ultraspherical

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Legendre

… ►See §18.18(vii) for Poisson kernels; these are special cases of bilateral generating functions.

16: 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).

Encodings:

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Asymptotic expansion of a class of integral transforms with algebraically dominated kernels. J. Math. Anal. Appl. 35 (2), pp. 405–433.

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External Links:

Document, MathReview (E. L. Koh), ZentralBlatt

Referenced by:

§2.5(ii).

Encodings:

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Bessel functions of matrix argument. Ann. of Math. (2) 61 (3), pp. 474–523.

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External Links:

FullText, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§35.1, §35.2, §35.3(ii), §35.5(i), §35.5(ii), §35.5(iii), §35.6(i), §35.6(ii), §35.6(iii), §35.6(iv), §35.7(i), §35.7(ii), §35.7(iv), §35.8(i), §35.8(ii), §35.8(iv), Ch.35, Notations P.

Encodings:

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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).

Encodings:

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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:

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