Bessel inequality

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11: 10.21 Zeros

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§10.21(v) Inequalities

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12: Bibliography M

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

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External Links:

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

Referenced by:

§10.21(iv).

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13: Errata

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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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14: 13.22 Zeros

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13.22.1 $${\varphi }_r=\frac{j_{2\mu ,r}^2}{4\kappa }+j_{2\mu ,r}O\left({\kappa }^{-\frac{3}{2}}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, ${\varphi }_r$: positive zeros and $j_{b,r}$: positive zero of Bessel

Referenced by:

§13.22, §13.22

Permalink:

http://dlmf.nist.gov/13.22.E1

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Annotations for §13.22 and Ch.13

►where $j_{2\mu ,r}$ is the $r$th positive zero of the Bessel function $J_{2\mu }\left(x\right)$ (§10.21(i)). …

15: 13.9 Zeros

… ►When and $b>0$ let ${\varphi }_r$, $r=1,2,3,\mathrm{\dots }$, be the positive zeros of $M(a,b,x)$ arranged in increasing order of magnitude, and let $j_{b-1,r}$ be the $r$th positive zero of the Bessel function $J_{b-1}\left(x\right)$ (§10.21(i)). … ►

13.9.8 $${\varphi }_r=\frac{j_{b-1,r}^2}{2b-4a}\left(1+\frac{2b(b-2)+j_{b-1,r}^2}{3{(2b-4a)}^2}\right)+O\left(\frac{1}{a^5}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $r=1,2,3,\mathrm{\dots }$ , ${\varphi }_r$: positive zeros and $j_{b,r}$: positive zero of Bessel

Referenced by:

§13.22, §13.9(i)

Permalink:

http://dlmf.nist.gov/13.9.E8

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Annotations for §13.9(i), §13.9 and Ch.13

… ►Inequalities for ${\varphi }_r$ are given in Gatteschi (1990), and identities involving infinite series of all of the complex zeros of $M(a,b,x)$ are given in Ahmed and Muldoon (1980). … ►Inequalities for the zeros of $U(a,b,x)$ are given in Gatteschi (1990). …

16: Bibliography E

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Some recent results on the zeros of Bessel functions and orthogonal polynomials. J. Comput. Appl. Math. 133 (1-2), pp. 65–83.

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External Links:

ISSN 0377-0427, Document, MathReview (N. Hayek Calil), ZentralBlatt

Referenced by:

§10.21(iv).

Encodings:

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Interlacing properties of the zeros of Bessel functions. Atti Sem. Mat. Fis. Univ. Modena XLII (2), pp. 525–529.

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External Links:

ISSN 0041-8986, MathReview (Evangelos Ifantis), ZentralBlatt

Referenced by:

§10.21(i).

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An upper bound for the zeros of the derivative of Bessel functions. Rend. Circ. Mat. Palermo (2) 46 (1), pp. 123–130.

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External Links:

ISSN 0009-725X, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.21(vii).

Encodings:

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Real orthogonalizing weights for Bessel polynomials. J. Comput. Appl. Math. 49 (1-3), pp. 51–57.

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External Links:

ISSN 0377-0427, Document, MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§18.34(ii).

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On an integral inequality. Proc. Roy. Soc. London Ser. A 357, pp. 271–288.

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External Links:

ISSN 0962-8444, FullText, MathReview, ZentralBlatt

Referenced by:

§10.37.

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17: 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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On the zeros of generalized Bessel polynomials. I. Nederl. Akad. Wetensch. Indag. Math. 84 (1), pp. 1–13.

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External Links:

ISSN 0019-3577, Document, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§10.49(ii), §10.58.

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On the zeros of generalized Bessel polynomials. II. Nederl. Akad. Wetensch. Indag. Math. 84 (1), pp. 14–25.

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External Links:

ISSN 0019-3577, Document, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§10.49(ii), §10.58.

Encodings:

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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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External Links:

Document

Referenced by:

§10.76(iii).

Encodings:

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Inequalities for extreme zeros of some classical orthogonal and $q$-orthogonal polynomials. Math. Model. Nat. Phenom. 8 (1), pp. 48–59.

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External Links:

ISSN 0973-5348, Document, FullText, MathReview (Ana Foulquié Moreno), ZentralBlatt

Referenced by:

§18.16(ii), §18.16(iv), §18.16(ii), §18.16(iv), §18.27(iv), §18.27(iv), §18.27(v), §18.27(v).

Encodings:

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18: Bibliography K

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The computation of the sums of negative even powers of roots of Bessel functions. Doklady Akad. Nauk SSSR (N.S.) 77, pp. 561–564.

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§10.21(xiii).

Encodings:

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Calculation of modified Bessel functions in a complex domain. Zh. Vychisl. Mat. i Mat. Fiz. 24 (5), pp. 650–664.

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Notes:

English translation in U.S.S.R. Computational Math. and Math. Phys. 24(3), pp. 15–24

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ISSN 0044-4669, Document, MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

§10.42, §10.74(iv).

Encodings:

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Some extensions of W. Gautschi’s inequalities for the gamma function. Math. Comp. 41 (164), pp. 607–611.

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External Links:

ISSN 0025-5718, FullText, MathReview (P. Anandani), ZentralBlatt

Referenced by:

§5.6(i).

Encodings:

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Jacobi polynomials, Bernstein-type inequalities and dispersion estimates for the discrete Laguerre operator. Adv. Math. 333, pp. 796–821.

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External Links:

ISSN 0001-8708, Document, Link, MathReview (Martin Axel Hallnäs)

Referenced by:

§18.14(i).

Encodings:

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Bessel Functions and their Applications. Analytical Methods and Special Functions, Vol. 8, Taylor & Francis Ltd., London-New York.

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Notes:

Translated from the Russian by E. V. Pankratiev

External Links:

ISBN 0-415-28130-X, MathReview (L. Debnath), ZentralBlatt

Referenced by:

§10.73(i), §10.73(i), §10.73(i).

Encodings:

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19: 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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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.

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A series of inequalities for Mills’s ratio. Acta Math. Sinica 25 (6), pp. 660–670.

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External Links:

MathReview (J. S. Huang), ZentralBlatt

Referenced by:

§7.8.

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20: Bibliography

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Monotonicity theorems and inequalities for the complete elliptic integrals. J. Comput. Appl. Math. 172 (2), pp. 289–312.

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External Links:

ISSN 0377-0427, Document, MathReview (Živorad Tomovski), ZentralBlatt

Referenced by:

§19.9(i).

Encodings:

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A harmonic mean inequality for the gamma function. J. Comput. Appl. Math. 87 (2), pp. 195–198.

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Notes:

For corrigendum see same journal v. 90 (1998), no. 2, p. 265

External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§5.6(i).

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On some inequalities for the incomplete gamma function. Math. Comp. 66 (218), pp. 771–778.

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External Links:

ISSN 0025-5718, Document, MathReview, ZentralBlatt

Referenced by:

§8.10.

Encodings:

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Gamma function inequalities. Numer. Algorithms 49 (1-4), pp. 53–84.

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External Links:

ISSN 1017-1398, Document, FullText, MathReview (Junesang Choi), ZentralBlatt

Referenced by:

§5.6(i), §5.6(i).

Encodings:

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Inequalities for elliptic integrals. Publ. Inst. Math. (Beograd) (N.S.) 37(51), pp. 61–63.

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External Links:

ISSN 0350-1302, MathReview, ZentralBlatt

Referenced by:

§19.9(i).

Encodings:

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