Bessel inequality

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21: Bibliography Q

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Some inequalities of the incomplete gamma and related functions. Z. Anal. Anwendungen 18 (3), pp. 793–799.

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External Links:

ISSN 0232-2064, FullText, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§8.10.

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A new lower bound in the second Kershaw’s double inequality. J. Comput. Appl. Math. 214 (2), pp. 610–616.

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ISSN 0377-0427, Document, FullText, MathReview (Mehdi Hassani), ZentralBlatt

Referenced by:

§5.6(i), §5.6(i).

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“Best possible” upper and lower bounds for the zeros of the Bessel function $J_{\nu }(x)$ . Trans. Amer. Math. Soc. 351 (7), pp. 2833–2859.

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External Links:

ISSN 0002-9947, FullText, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.21(vii).

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22: Bibliography B

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New tables of Bessel functions of complex argument. Comput. Math. Math. Phys. 37 (12), pp. 1480–1482.

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Notes:

Russian original in Zh. Vychisl. Mat. i Mat. Fiz. 37(1997), no. 12, 1526–1528

External Links:

ZentralBlatt

Referenced by:

§10.75(i).

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Automatic computation of zeros of Bessel functions and other special functions. SIAM J. Sci. Comput. 21 (4), pp. 1458–1464.

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External Links:

ISSN 1064-8275, Document, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.74(vi), §3.8(iii).

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Inequalities for the perimeter of an ellipse. J. Math. Anal. Appl. 260 (2), pp. 295–306.

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ISSN 0022-247X, Document, MathReview (Matti Vuorinen), ZentralBlatt

Referenced by:

§19.9(i).

Encodings:

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Introduction to Bessel Functions. Dover Publications Inc., New York.

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Notes:

Unaltered republication of the 1938 edition, published by Longmans, Green and Co., London-New York.

External Links:

MathReview, ZentralBlatt

Referenced by:

§10.73(iii).

Encodings:

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A Dictionary of Inequalities. Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 97, Longman, Harlow.

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External Links:

ISBN 0-8493-0634-5, MathReview (Lucio Renato Berrone), ZentralBlatt

Referenced by:

§4.18, §4.5(i), §4.5(ii).

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23: 26.10 Integer Partitions: Other Restrictions

… ►Note that $p({𝒟}^{\prime }3,n)\le p(𝒟3,n)$, with strict inequality for $n\ge 9$. It is known that for $k>3$, $p(𝒟k,n)\ge p(\in A_{1,k+3},n)$, with strict inequality for $n$ sufficiently large, provided that $k=2^m-1,m=3,4,5$, or $k\ge 32$; see Yee (2004). … ►

§26.10(vi) Bessel-Function Expansion

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26.10.17 $$p(𝒟,n)=\pi \sum _{k=1}^{\mathrm{\infty }}\frac{A_{2k-1}(n)}{(2k-1)\sqrt{24n+1}}{I}_1\left(\frac{\pi }{2k-1}\sqrt{\frac{24n+1}{72}}\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $p(\mathrm{condition},n)$: restricted number of partitions of $n$ , $k$: nonnegative integer, $n$: nonnegative integer and $A_k(n)$: coefficient

A&S Ref:

24.2.2

Permalink:

http://dlmf.nist.gov/26.10.E17

Encodings:

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See also:

Annotations for §26.10(vi), §26.10 and Ch.26

►where $I_1\left(x\right)$ is the modified Bessel function (§10.25(ii)), and …

24: 18.39 Applications in the Physical Sciences

… ►The functions ${\varphi }_n$ are expressed in terms of Romanovski–Bessel polynomials, or Laguerre polynomials by (18.34.7_1). The finite system of functions ${\psi }_n$ is orthonormal in $L^2(ℝ,dx)$, see (18.34.7_3). … ►Note that violation of the Favard inequality, $l+1+(2Z/s)>0,$ possible when , results in a zero or negative weight function. … ►See Yamani and Fishman (1975) for $L^2$ for expansions of both the regular and irregular spherical Bessel functions, which are the Pollaczeks with $a=Z=0$, and Coulomb functions for fixed $l$, Broad and Reinhardt (1976) for a many particle example, and the overview of Alhaidari et al. (2008). … ►For applications and an extension of the Szegő–Szász inequality (18.14.20) for Legendre polynomials ($\alpha =\beta =0$) to obtain global bounds on the variation of the phase of an elastic scattering amplitude, see Cornille and Martin (1972, 1974). …

25: 11.4 Basic Properties

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11.4.3 $${𝐇}_{-n-\frac{1}{2}}\left(z\right)={(-1)}^n{J}_{n+\frac{1}{2}}\left(z\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, ${𝐇}_{\nu }\left(z\right)$: Struve function, $z$: complex variable and $n$: integer order

A&S Ref:

12.1.15

Referenced by:

§11.4(i)

Permalink:

http://dlmf.nist.gov/11.4.E3

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See also:

Annotations for §11.4(i), §11.4 and Ch.11

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11.4.4 $${𝐋}_{-n-\frac{1}{2}}\left(z\right)=I_{n+\frac{1}{2}}\left(z\right).$$

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Symbols:

$I_{\nu }\left(z\right)$: modified Bessel function of the first kind, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $z$: complex variable and $n$: integer order

A&S Ref:

12.2.10

Referenced by:

§11.4(i)

Permalink:

http://dlmf.nist.gov/11.4.E4

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See also:

Annotations for §11.4(i), §11.4 and Ch.11

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§11.4(ii) Inequalities

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§11.4(iv) Expansions in Series of Bessel Functions

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11.4.22 $${𝐇}_1\left(z\right)=\frac{2}{\pi }(1-J_0\left(z\right))+\frac{4}{\pi }\sum _{k=1}^{\mathrm{\infty }}\frac{J_{2k}\left(z\right)}{4k^2-1}=4\sum _{k=0}^{\mathrm{\infty }}{J}_{2k+\frac{1}{2}}\left(\frac{1}{2}z\right)J_{2k+\frac{3}{2}}\left(\frac{1}{2}z\right).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, ${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable and $k$: nonnegative integer

A&S Ref:

12.1.20 (First part)

Permalink:

http://dlmf.nist.gov/11.4.E22

Encodings:

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See also:

Annotations for §11.4(iv), §11.4 and Ch.11

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26: 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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Inequalities. 2nd edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge.

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Notes:

Reprint of the 1952 edition and reprinted in 1988.

External Links:

ISBN 0-521-35880-9, MathReview, ZentralBlatt

Referenced by:

§1.2(iv), §1.4(viii), §1.7(i), §1.7(ii), §1.7(iii), §1.7(iv), §4.5(i), §4.5(ii).

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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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External Links:

ISSN 0168-9274, Document, MathReview, ZentralBlatt

Referenced by:

§10.60(iii), §6.10(ii), §6.15.

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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.

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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).

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27: Bibliography C

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An algorithm for the Fourier-Bessel transform. Comput. Phys. Comm. 23 (4), pp. 343–353.

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External Links:

ISSN 0010-4655, Document, MathReview

Referenced by:

§10.74(vii).

Encodings:

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Some inequalities for hypergeometric functions. Proc. Amer. Math. Soc. 17 (1), pp. 32–39.

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External Links:

FullText, MathReview (Gabor Szegő), ZentralBlatt

Referenced by:

§19.24(i), §19.24(i), §19.24(ii), §19.9(i), §19.9(ii).

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Inequalities for a symmetric elliptic integral. Proc. Amer. Math. Soc. 25 (3), pp. 698–703.

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External Links:

FullText, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§19.24(ii), §19.9(ii).

Encodings:

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A Bernstein-type inequality for the Jacobi polynomial. Proc. Amer. Math. Soc. 121 (3), pp. 703–709.

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External Links:

ISSN 0002-9939, FullText, MathReview (Joaquin Bustoz), ZentralBlatt

Referenced by:

§18.14(i).

Encodings:

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The analyticity of cross-product Bessel function zeros. Proc. Cambridge Philos. Soc. 62, pp. 215–226.

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External Links:

MathReview (R. G. Langebartel), ZentralBlatt

Referenced by:

§10.21(x).

Encodings:

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