Bessel equation

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51: 28.34 Methods of Computation

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§28.34(i) Characteristic Exponents

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§28.34(ii) Eigenvalues

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(c)

Methods described in §3.7(iv) applied to the differential equation (28.2.1) with the conditions (28.2.5) and (28.2.16).

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§28.34(iii) Floquet Solutions

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(b)

Direct numerical integration (§3.7) of the differential equation (28.20.1) for moderate values of the parameters.

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52: 10.12 Generating Function and Associated Series

§10.12 Generating Function and Associated Series

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10.12.4

1=J_{0}\left(z\right)+2J_{2}\left(z\right)+2J_{4}\left(z\right)+2J_{6}\left(z% \right)+\dotsb,

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind and $z$ : complex variable
A&S Ref:: 9.1.46
Referenced by:: §10.74(iv)
Permalink:: http://dlmf.nist.gov/10.12.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §10.12 and Ch.10

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\cos z=J_{0}\left(z\right)-2J_{2}\left(z\right)+2J_{4}\left(z\right)-2J_{6}% \left(z\right)+\dotsb,

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\sin z=2J_{1}\left(z\right)-2J_{3}\left(z\right)+2J_{5}\left(z\right)-\dotsb,

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\tfrac{1}{2}z\cos z=J_{1}\left(z\right)-9J_{3}\left(z\right)+25J_{5}\left(z% \right)-49J_{7}\left(z\right)+\dotsb,

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53: Bibliography V

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J. Van Deun and R. Cools (2008) Integrating products of Bessel functions with an additional exponential or rational factor. Comput. Phys. Comm. 178 (8), pp. 578–590.

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Notes:: Associated computer program available online
External Links:: ISSN 0010-4655, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §10.74(vii), §10.74(vii).
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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G. Vedeler (1950) A Mathieu equation for ships rolling among waves. I, II. Norske Vid. Selsk. Forh., Trondheim 22 (25–26), pp. 113–123.

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External Links:: MathReview (J. V. Wehausen), ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

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A. P. Vorob’ev (1965) On the rational solutions of the second Painlevé equation. Differ. Uravn. 1 (1), pp. 79–81 (Russian).

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Notes:: In Russian. English translation: Differential Equations 1(1), pp. 58–59.
External Links:: MathReview, ZentralBlatt
Referenced by:: §32.8(ii).
Encodings:: BibTeX

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M. N. Vrahatis, T. N. Grapsa, O. Ragos, and F. A. Zafiropoulos (1997a) On the localization and computation of zeros of Bessel functions. Z. Angew. Math. Mech. 77 (6), pp. 467–475.

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External Links:: ISSN 0044-2267, Document, MathReview, ZentralBlatt
Referenced by:: §10.74(vi).
Encodings:: BibTeX

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54: Bibliography N

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A. Nakamura (1996) Toda equation and its solutions in special functions. J. Phys. Soc. Japan 65 (6), pp. 1589–1597.

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External Links:: ISSN 0031-9015, Document, MathReview, ZentralBlatt
Referenced by:: §18.38(ii).
Encodings:: BibTeX

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G. Nemes (2017b) Error Bounds for the Large-Argument Asymptotic Expansions of the Hankel and Bessel Functions. Acta Appl. Math. 150, pp. 141–177.

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External Links:: ISSN 0167-8019, Document, MathReview Entry, ZentralBlatt
Referenced by:: (9.7.16), (9.7.17), §9.7(iii), §9.7(iv).
Encodings:: BibTeX

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E. Neuman (2004) Inequalities involving Bessel functions of the first kind. JIPAM. J. Inequal. Pure Appl. Math. 5 (4), pp. Article 94, 4 pp. (electronic).

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External Links:: ISSN 1443-5756, FullText, MathReview, ZentralBlatt
Referenced by:: §10.14.
Encodings:: BibTeX

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J. N. Newman (1984) Approximations for the Bessel and Struve functions. Math. Comp. 43 (168), pp. 551–556.

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External Links:: ISSN 0025-5718, FullText, MathReview (S. Conde), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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M. Newman (1967) Solving equations exactly. J. Res. Nat. Bur. Standards Sect. B 71B, pp. 171–179.

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External Links:: MathReview, ZentralBlatt
Referenced by:: §27.15.
Encodings:: BibTeX

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55: 36.2 Catastrophes and Canonical Integrals

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36.2.28

\Psi^{(\mathrm{E})}\left(0,0,z\right)=\overline{\Psi^{(\mathrm{E})}\left(0,0,-% z\right)}\\ =2\pi\sqrt{\frac{\pi z}{27}}\exp\left(\frac{2}{27}iz^{3}\right)\*\left(J_{-1/6% }\left(\frac{2}{27}z^{3}\right)+iJ_{1/6}\left(\frac{2}{27}z^{3}\right)\right),

z\geq 0

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$ : complex conjugate, $\Psi^{(\mathrm{E})}\left(\NVar{\mathbf{x}}\right)$ : elliptic umbilic canonical integral function, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit and $z$ : real parameter
Referenced by:: Erratum (V1.0.5) for Chapters 8, 20, 36
Permalink:: http://dlmf.nist.gov/36.2.E28
Encodings:: pMML, png, TeX
Addition (effective with 1.0.5):: This equation has been added. For the proof see Berry and Howls (2010).
See also:: Annotations for §36.2(iv), §36.2 and Ch.36

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56: 10.39 Relations to Other Functions

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Elementary Functions

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Parabolic Cylinder Functions

… ►Principal values on each side of these equations correspond. … ►

Confluent Hypergeometric Functions

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Generalized Hypergeometric Functions and Hypergeometric Function

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57: 28.8 Asymptotic Expansions for Large $q$

§28.8 Asymptotic Expansions for Large $q$

… ►Barrett (1981) supplies asymptotic approximations for numerically satisfactory pairs of solutions of both Mathieu’s equation (28.2.1) and the modified Mathieu equation (28.20.1). …The approximants are elementary functions, Airy functions, Bessel functions, and parabolic cylinder functions; compare §2.8. It is stated that corresponding uniform approximations can be obtained for other solutions, including the eigensolutions, of the differential equations by application of the results, but these approximations are not included. … ►Dunster (1994a) supplies uniform asymptotic approximations for numerically satisfactory pairs of solutions of Mathieu’s equation (28.2.1). …

58: 18.16 Zeros

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18.16.2

\theta_{n,m}^{(-\frac{1}{2},\frac{1}{2})}=\frac{(m-\tfrac{1}{2})\pi}{n+\tfrac{% 1}{2}}\leq\theta_{n,m}^{(\alpha,\beta)}\leq\frac{m\pi}{n+\tfrac{1}{2}}=\theta_% {n,m}^{(\frac{1}{2},-\frac{1}{2})},

\alpha,\beta\in[-\tfrac{1}{2},\tfrac{1}{2}]

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $[\NVar{a},\NVar{b}]$ : closed interval, $\in$ : element of, $(\NVar{a},\NVar{b})$ : open interval, $m$ : nonnegative integer, $n$ : nonnegative integer and $\theta_{n,m}$ : zeros
Referenced by:: §18.16(iii), Erratum (V1.2.0) for Equations (18.16.2), (18.16.3)
Permalink:: http://dlmf.nist.gov/18.16.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §18.16(ii), §18.16(ii), §18.16 and Ch.18

… ►Let

j_{\alpha,m}

be the

m

th positive zero of the Bessel function

J_{\alpha}\left(x\right)

(§10.21(i)). Then … ►

18.16.12

(n+2)x_{n,1}\geq\left(n-1-\sqrt{n^{2}+(n+2)(\alpha+1)}\right)^{2}-1,

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Symbols:: $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.16(iv), Erratum (V1.0.5) for Equations (18.16.12), (18.16.13), Erratum (V1.2.0) for Equations (18.16.12), (18.16.13)
Permalink:: http://dlmf.nist.gov/18.16.E12
Encodings:: pMML, png, TeX
Addition (effective with 1.0.5):: This bound, taken from Dimitrov and Nikolov (2010), replaces the weaker bound $x_{n,1}>2n+\alpha-2-(1+4(n-1)(n+\alpha-1))^{\frac{1}{2}}$ , which had been taken from Ismail and Li (1992).
See also:: Annotations for §18.16(iv), §18.16(iv), §18.16 and Ch.18

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18.16.13

(n+2)x_{n,n}\leq\left(n-1+\sqrt{n^{2}+(n+2)(\alpha+1)}\right)^{2}-1.

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Symbols:: $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.16(iv), Erratum (V1.0.5) for Equations (18.16.12), (18.16.13), Erratum (V1.2.0) for Equations (18.16.12), (18.16.13)
Permalink:: http://dlmf.nist.gov/18.16.E13
Encodings:: pMML, png, TeX
Modification (effective with 1.0.5):: This bound, taken from Dimitrov and Nikolov (2010), replaces the weaker bound $x_{n,n}<2n+\alpha-2+(1+4(n-1)(n+\alpha-1))^{\frac{1}{2}}$ , which had been taken from Ismail and Li (1992).
See also:: Annotations for §18.16(iv), §18.16(iv), §18.16 and Ch.18

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59: Bibliography H

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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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N. J. Hitchin (1995) Poncelet Polygons and the Painlevé Equations. In Geometry and Analysis (Bombay, 1992), Ramanan (Ed.), pp. 151–185.

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External Links:: MathReview (Henrik Pedersen), ZentralBlatt
Referenced by:: §32.9(iii).
Encodings:: BibTeX

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H. Hochstadt (1963) Estimates of the stability intervals for Hill’s equation. Proc. Amer. Math. Soc. 14 (6), pp. 930–932.

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External Links:: FullText, MathReview (P. Ungar), ZentralBlatt
Referenced by:: §28.29(iii).
Encodings:: BibTeX

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H. Hochstadt (1964) Differential Equations: A Modern Approach. Holt, Rinehart and Winston, New York.

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Notes:: Reprinted by Dover in 1975.
External Links:: MathReview (Z. Opial), ZentralBlatt
Referenced by:: §29.3(vi).
Encodings:: BibTeX

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60: Mathematical Introduction

… ►This is because

\mathbf{F}

is akin to the notation used for Bessel functions (§10.2(ii)), inasmuch as

\mathbf{F}

is an entire function of each of its parameters

a

b

, and

c

: this results in fewer restrictions and simpler equations. …