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1: 10.73 Physical Applications

… ►Bessel functions of the first kind, $J_n\left(x\right)$, arise naturally in applications having cylindrical symmetry in which the physics is described either by Laplace’s equation ${\nabla }^{2}V=0$, or by the Helmholtz equation $({\nabla }^2+k^2)\psi =0$. … ►In cylindrical coordinates $r$, $\varphi$, $z$, (§1.5(ii) we have … ►See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25). … ►Bessel functions enter in the study of the scattering of light and other electromagnetic radiation, not only from cylindrical surfaces but also in the statistical analysis involved in scattering from rough surfaces. … ►On separation of variables into cylindrical coordinates, the Bessel functions $J_n\left(x\right)$, and modified Bessel functions $I_n\left(x\right)$ and $K_n\left(x\right)$, all appear. …

2: 1.5 Calculus of Two or More Variables

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§1.5(ii) Coordinate Systems

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Polar Coordinates

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Cylindrical Coordinates

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Spherical Coordinates

… ►For applications and other coordinate systems see §§12.17, 14.19(i), 14.30(iv), 28.32, 29.18, 30.13, 30.14. …

3: Bibliography K

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On multiple zeros of derivatives of Bessel’s cylindrical functions. Dokl. Akad. Nauk SSSR 288 (2), pp. 285–288 (Russian).

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Notes:

In Russian. English translation: Soviet Math. Dokl. 33(3), 650–653, (1986).

External Links:

ISSN 0002-3264, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.21(i), §10.74(vi).

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On the calculation of the multiple complex roots of the derivatives of cylindrical Bessel functions. Zh. Vychisl. Mat. i Mat. Fiz. 27 (11), pp. 1628–1639, 1758.

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Notes:

English translation in U.S.S.R. Computational Math. and Math. Phys. 27(6), pp. 18–25

External Links:

ISSN 0044-4669, Document, MathReview (Hans-Jürgen Albrand), ZentralBlatt

Referenced by:

§10.21(ix), §10.74(vi), 5th item.

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Multiple complex zeros of derivatives of the cylindrical Bessel functions. Dokl. Akad. Nauk SSSR 299 (3), pp. 614–618 (Russian).

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Notes:

In Russian. English translation: Soviet Phys. Dokl. 33(3),196–198, (1988).

External Links:

ISSN 0002-3264, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.21(ix), §10.74(vi).

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Algorithm 877: A subroutine package for cylindrical functions of complex order and nonnegative argument. ACM Trans. Math. Software 34 (4), pp. Art. 22, 21.

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External Links:

ISSN 0098-3500, Document, MathReview Entry, ZentralBlatt

Referenced by:

1st item.

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Algorithm 912: a module for calculating cylindrical functions of complex order and complex argument. ACM Trans. Math. Software 37 (4), pp. Art. 47, 25.

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External Links:

Document, ZentralBlatt

Referenced by:

2nd item, §10.77(viii).

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4: Bibliography M

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Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

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Notes:

Includes Fortran code, maximum accuracy 20S.

External Links:

ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§6.13, 4th item, §6.21(ii).

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Lie theory and separation of variables. I: Parabolic cylinder coordinates. SIAM J. Math. Anal. 5 (4), pp. 626–643.

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External Links:

Document, MathReview (G. R. Allcock), ZentralBlatt

Referenced by:

§12.13(ii), §12.17.

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The $q$-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

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External Links:

ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.27(v).

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Calculation of zeros of cylindrical functions and their derivatives. Vestnik Moskov. Univ. Ser. XV Vychisl. Mat. Kibernet. (2), pp. 63–64, 71 (Russian).

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Notes:

In Russian

External Links:

ISSN 0137-0782, MathReview, ZentralBlatt

Referenced by:

§10.74(vi).

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Field Theory Handbook. Including Coordinate Systems, Differential Equations and Their Solutions. 2nd edition, Springer-Verlag, Berlin.

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Notes:

Reprinted in 1988.

External Links:

ISBN 3-540-18430-9, MathReview

Referenced by:

Table 28.1.1.

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5: 36.5 Stokes Sets

… ►For $z\ne 0$, the Stokes set is expressed in terms of scaled coordinates … ►

36.5.7 $$X=\frac{9}{20}+20u^4-\frac{Y^2}{20u^2}+6u^2\mathrm{sign}\left(z\right),$$

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Symbols:

$\mathrm{sign}x$: sign of, $z$: real parameter, $X$: scaled coordinate and $Y$: scaled coordinate

Permalink:

http://dlmf.nist.gov/36.5.E7

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See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

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36.5.10 $$160u^6+40u^4=Y^2.$$

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Symbols:

$Y$: scaled coordinate

Permalink:

http://dlmf.nist.gov/36.5.E10

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Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

… ►With coordinates … ►

36.5.17 $$Y_{\mathrm{S}}(X)=Y(u,|X|),$$

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Symbols:

$X$: scaled coordinate

Permalink:

http://dlmf.nist.gov/36.5.E17

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See also:

Annotations for §36.5(iii), §36.5(iii), §36.5 and Ch.36

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6: Bibliography H

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Certain sums that contain cylindrical functions. Bul. Akad. Štiince RSS Moldoven. 1972 (3), pp. 75–77, 94 (Russian).

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External Links:

MathReview (D. P. Gupta)

Referenced by:

§7.14(iii).

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Sums with cylindrical functions that reduce to the probability function and to related functions. Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 80–84, 95 (Russian).

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External Links:

MathReview (M. Lawrence Glasser)

Referenced by:

§7.14(iii).

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The combination of $R$-matrix and complex coordinate methods: Application to the diamagnetic Rydberg spectra of Ba and Sr. J. Phys. B 26 (12), pp. 1775–1790.

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External Links:

ISSN 0953-4075, Document, MathReview

Referenced by:

4th item.

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7: 20 Theta Functions

Chapter 20 Theta Functions

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8: 29.18 Mathematical Applications

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§29.18(i) Sphero-Conal Coordinates

… ►when transformed to sphero-conal coordinates $r,\beta ,\gamma$: … ►

29.18.4 $$u(r,\beta ,\gamma )=u_1(r)u_2(\beta )u_3(\gamma ),$$

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Symbols:

$u$: solution, $r$: sphero-conal coordinate, $\beta$: sphero-conal coordinate, $\gamma$: sphero-conal coordinate and $u_j$: solutions

Permalink:

http://dlmf.nist.gov/29.18.E4

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See also:

Annotations for §29.18(i), §29.18 and Ch.29

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§29.18(ii) Ellipsoidal Coordinates

►The wave equation (29.18.1), when transformed to ellipsoidal coordinates $\alpha ,\beta ,\gamma$: …

9: 12.17 Physical Applications

§12.17 Physical Applications

… ►in Cartesian coordinates $x,y,z$ of three-dimensional space (§1.5(ii)). By using instead coordinates of the parabolic cylinder $\xi ,\eta ,\zeta$, defined by … ►In a similar manner coordinates of the paraboloid of revolution transform the Helmholtz equation into equations related to the differential equations considered in this chapter. … …

10: Bibliography S

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Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.

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External Links:

ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§22.19(i), §22.20(vi).

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Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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External Links:

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

Referenced by:

§28.8(iv).

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Inequalities involving cylindrical functions of nearly equal argument and order. Proc. Amer. Math. Soc. 5 (3), pp. 337–344.

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External Links:

ISSN 0002-9939, FullText, MathReview (N. D. Kazarinoff), ZentralBlatt

Referenced by:

§10.14.

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Some results on the zeros of cylindrical functions and of their derivatives. Rend. Sem. Mat. Univ. Politec. Torino 38 (1), pp. 67–85 (Italian. English summary).

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External Links:

MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§10.21(vii).

Encodings:

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A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.

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Notes:

For software see John Stembridge’s home page. The SF package contains Maple software for zonal, Jack, and Macdonald polynomials.

External Links:

Home Page, ISSN 0747-7171, Document, MathReview, ZentralBlatt

Referenced by:

3rd item.

Encodings:

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