products of Bessel functions

SearchAdvancedHelp

(0.010 seconds)

11—20 of 48 matching pages

11: 10.43 Integrals

… ► … ►For infinite integrals of triple products of modified and unmodified Bessel functions, see Gervois and Navelet (1984, 1985a, 1985b, 1986a, 1986b). …

12: 10.32 Integral Representations

… ►

§10.32(iii) Products

… ►For collections of integral representations of modified Bessel functions, or products of modified Bessel functions, see Erdélyi et al. (1953b, §§7.3, 7.12, and 7.14.2), Erdélyi et al. (1954a, pp. 48–60, 105–115, 276–285, and 357–359), Gröbner and Hofreiter (1950, pp. 193–194), Magnus et al. (1966, §3.7), Marichev (1983, pp. 191–216), and Watson (1944, Chapters 6, 12, and 13).

13: Bibliography M

… ►

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

ⓘ

External Links:

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

Referenced by:

§10.21(x).

Encodings:

BibTeX

… ►

On the evaluation of the integral over the product of two spherical Bessel functions. J. Math. Phys. 32 (3), pp. 642–648.

ⓘ

External Links:

ISSN 0022-2488, Document, MathReview (S. K. Chatterjea), ZentralBlatt

Referenced by:

§1.17(ii), §10.59.

Encodings:

BibTeX

… ►

Analytic expressions for integrals of products of spherical Bessel functions. J. Phys. A 24 (7), pp. 1435–1453.

ⓘ

External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§10.59.

Encodings:

BibTeX

… ►

On the zeros of a cross-product of Bessel functions of different orders. Z. Angew. Math. Mech. 59 (6), pp. 272–273.

ⓘ

External Links:

ISSN 0044-2267, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§10.21(x).

Encodings:

BibTeX

…

14: Bibliography V

… ►

Integrating products of Bessel functions with an additional exponential or rational factor. Comput. Phys. Comm. 178 (8), pp. 578–590.

ⓘ

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

…

15: 10.59 Integrals

… ►For an integral representation of the Dirac delta in terms of a product of spherical Bessel functions of the first kind see §1.17(ii), and for a generalization see Maximon (1991). …

16: 28.28 Integrals, Integral Representations, and Integral Equations

… ►

§28.28(ii) Integrals of Products with Bessel Functions

… ►

28.28.23 $$\frac{2}{\pi }{\int }_0^{\pi }{𝒞}_{2\mathrm{\ell }+2}^{(j)}(2hR)\sin \left((2\mathrm{\ell }+2)\varphi \right){\mathrm{se}}_{2m+2}(t,h^2)dt={(-1)}^{\mathrm{\ell }+m}{B}_{2\mathrm{\ell }+2}^{2m+2}(h^2){\mathrm{Ms}}_{2m+2}^{(j)}(z,h).$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $\sin z$: sine function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions, $\varphi (z,t)$: function, $R(z,t)$: function and $B_m(q)$: Fourier coefficient

A&S Ref:

20.7.28 (in different form and only for $\mathrm{\ell }=0$)

Permalink:

http://dlmf.nist.gov/28.28.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(ii), §28.28 and Ch.28

…

17: 11.10 Anger–Weber Functions

… ►

§11.10(viii) Expansions in Series of Products of Bessel Functions

…

18: Bibliography L

… ►

Numerical evaluation of integrals containing a spherical Bessel function by product integration. J. Math. Phys. 22 (7), pp. 1399–1413.

ⓘ

External Links:

ISSN 0022-2488, Document, MathReview (C. W. Clenshaw), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

… ►

A note on the computation of integrals involving products of trigonometric and Bessel functions. Math. Comp. 27 (124), pp. 871–872.

ⓘ

External Links:

ISSN 0025-5718, FullText, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

… ►

Evaluating infinite integrals involving products of Bessel functions of arbitrary order. J. Comput. Appl. Math. 64 (3), pp. 269–282.

ⓘ

External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

…

19: 10.21 Zeros

… ►

§10.21(iii) Infinite Products

►

10.21.15 $$J_{\nu }\left(z\right)=\frac{{(\frac{1}{2}z)}^{\nu }}{\mathrm{\Gamma }\left(\nu +1\right)}\prod _{k=1}^{\mathrm{\infty }}\left(1-\frac{z^2}{j_{\nu ,k}^2}\right),$$ $\nu \ge 0$,

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $j_{\nu ,m}$: zeros of the Bessel function $J_{\nu }\left(x\right)$ , $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.5.10

Referenced by:

§10.21(iii)

Permalink:

http://dlmf.nist.gov/10.21.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §10.21(iii), §10.21 and Ch.10

►

10.21.16 $$J_{\nu }^{\prime }\left(z\right)=\frac{{(\frac{1}{2}z)}^{\nu -1}}{2\mathrm{\Gamma }\left(\nu \right)}\prod _{k=1}^{\mathrm{\infty }}\left(1-\frac{z^2}{j_{\nu ,k}^{\prime }{}_{}{}^2}\right),$$ $\nu >0$.

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $j_{\nu ,m}^{\prime }$: zeros of the Bessel function derivative $J_{\nu }^{\prime }\left(x\right)$ , $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.5.11

Referenced by:

§10.21(iii)

Permalink:

http://dlmf.nist.gov/10.21.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §10.21(iii), §10.21 and Ch.10

… ►

§10.21(x) Cross-Products

… ►For information on the zeros of the derivatives of Riccati–Bessel functions, and also on zeros of their cross-products, see Boyer (1969). …

20: 10.40 Asymptotic Expansions for Large Argument

… ►

Products

…