Hankel integrals

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21: Bibliography C

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Determination of $\nu$-zeros of Hankel functions. Comput. Phys. Comm. 32 (3), pp. 333–339.

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Notes:

Double-precision Fortran, maximum accuracy 7S.

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ISSN 0010-4655, Document, Code

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1st item.

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A quadrature formula for the Hankel transform. Numer. Algorithms 9 (2), pp. 343–354.

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External Links:

ISSN 1017-1398, Document, MathReview (V. I. Polovinkin), ZentralBlatt

Referenced by:

§10.74(vii).

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Numerical integration of related Hankel transforms by quadrature and continued fraction expansion. Geophysics 48 (12), pp. 1671–1686.

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Notes:

Fortran code.

External Links:

ISSN 0098-3500, Document

Referenced by:

§10.77(ix).

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Optimized fast Hankel transform filters. Geophysical Prospecting 38 (5), pp. 545–568.

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External Links:

Document

Referenced by:

§10.74(vii).

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Computation of Hankel transforms. SIAM Rev. 14 (2), pp. 278–285.

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ISSN 1095-7200, Document, MathReview (R. D. Riess), ZentralBlatt

Referenced by:

§10.74(vii).

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22: Bibliography W

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Some useful integrals of $\mathrm{Si}(x),$ $\mathrm{Ci}(x)$ and related integrals. Optica Acta 14 (3), pp. 317–322.

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External Links:

Document, MathReview (S. D. Bajpai)

Referenced by:

§6.14(iii), §6.7(iv).

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Algorithm 794: Numerical Hankel transform by the Fortran program HANKEL. ACM Trans. Math. Software 25 (2), pp. 240–250.

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Notes:

Double-precision Fortran, maximum accuracy 10S.

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ISSN 0098-3500, Document, GAMS (module 794; package TOMS), ZentralBlatt

Referenced by:

7th item, 10th item.

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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).

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Error bounds for asymptotic expansions of Hankel transforms. SIAM J. Math. Anal. 7 (6), pp. 799–808.

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ISSN 1095-7154, Document, MathReview (John S. Lew), ZentralBlatt

Referenced by:

§10.22(v).

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Asymptotic expansions of Hankel transforms of functions with logarithmic singularities. Comput. Math. Appl. 3 (4), pp. 271–286.

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ISSN 0886-9561, Document, MathReview, ZentralBlatt

Referenced by:

§10.22(v).

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23: 18.2 General Orthogonal Polynomials

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18.2.1 $${\int }_a^b{p}_n(x)p_m(x)w(x)dx=0,$$ $n\ne m$.

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $p_n(x)$: polynomial of degree $n$ , $w(x)$: weight function, $m$: nonnegative integer, $n$: nonnegative integer, $a$: interval endpoint, $b$: interval endpoint and $x$: real variable

A&S Ref:

22.1.1

Referenced by:

§18.2(i), §18.2(iii), §18.2(viii), §18.5(iii)

Permalink:

http://dlmf.nist.gov/18.2.E1

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Annotations for §18.2(i), §18.2(i), §18.2 and Ch.18

… ►More generally than (18.2.1)–(18.2.3), $w(x)dx$ may be replaced in (18.2.1) by $d\mu (x)$, where the measure $\mu$ is the Lebesgue–Stieltjes measure ${\mu }_{\alpha }$ corresponding to a bounded nondecreasing function $\alpha$ on the closure of $(a,b)$ with an infinite number of points of increase, and such that for all $n$. … ►

§18.2(ix) Moments

… ►The Hankel determinant ${\mathrm{\Delta }}_n$ of order $n$ is defined by ${\mathrm{\Delta }}_0=1$ and …It is to be noted that, although formally correct, the results of (18.2.30) are of little utility for numerical work, as Hankel determinants are notoriously ill-conditioned. …

24: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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25: Bibliography H

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Une $q$-intégrale de Selberg et Askey. SIAM J. Math. Anal. 19 (6), pp. 1475–1489.

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External Links:

ISSN 0036-1410, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§17.13.

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Certain integrals that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1975 (2), pp. 86–88, 95 (Russian).

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External Links:

MathReview (C. Tanaka)

Referenced by:

§7.14(iii).

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Fast Hankel transform algorithm. IEEE Trans. Acoust. Speech Signal Process. 32 (3), pp. 666–671.

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External Links:

ISSN 0096-3518, FullText, ZentralBlatt

Referenced by:

§10.74(vii).

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Tables of the Modified Hankel Functions of Order One-Third and of their Derivatives. Harvard University Press, Cambridge, MA.

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

2nd item.

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Error bounds for asymptotic approximations of zeros of Hankel functions occurring in diffraction problems. J. Mathematical Phys. 11 (8), pp. 2501–2504.

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External Links:

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

Referenced by:

§10.21(xiv).

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26: 9.17 Methods of Computation

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§9.17(iii) Integral Representations

►Among the integral representations of the Airy functions the Stieltjes transform (9.10.18) furnishes a way of computing $\mathrm{Ai}\left(z\right)$ in the complex plane, once values of this function can be generated on the positive real axis. … ►In the first method the integration path for the contour integral (9.5.4) is deformed to coincide with paths of steepest descent (§2.4(iv)). …The second method is to apply generalized Gauss–Laguerre quadrature (§3.5(v)) to the integral (9.5.8). … ►In consequence of §9.6(i), algorithms for generating Bessel functions, Hankel functions, and modified Bessel functions (§10.74) can also be applied to $\mathrm{Ai}\left(z\right)$, $\mathrm{Bi}\left(z\right)$, and their derivatives. …

27: Bibliography

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Lectures on Integral Transforms. Translations of Mathematical Monographs, Vol. 70, American Mathematical Society, Providence, RI.

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Notes:

Translated from the Russian by H. H. McFaden

External Links:

ISBN 0-8218-4524-1, MathReview, ZentralBlatt

Referenced by:

§10.22(v).

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Integrals involving Airy functions. J. Phys. A 19 (13), pp. 2663–2665.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

(9.11.11), (9.11.12), (9.11.13), (9.11.14), §9.11(iv).

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Integrals involving Bickley and Bessel functions in radiative transfer, and generalized exponential integral functions. J. Heat Transfer 118 (3), pp. 789–792.

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External Links:

Document

Referenced by:

§10.73(iv), §8.24(iii).

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Computation of exponential integrals. ACM Trans. Math. Software 6 (3), pp. 365–377.

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External Links:

ISSN 0098-3500, Document, MathReview (M. M. Chawla), ZentralBlatt

Referenced by:

§8.25(v).

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Algorithm 588. Fast Hankel transforms using related and lagged convolutions. ACM Trans. Math. Software 8 (4), pp. 369–370.

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Notes:

Single-precision Fortran, maximum accuracy 8S.

External Links:

ISSN 0098-3500, Document, GAMS (module 588; package TOMS)

Referenced by:

2nd item.

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28: 10.77 Software

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§10.77(v) Bessel Functions–Real Order and Complex Argument (including Hankel Functions)

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§10.77(ix) Integrals of Bessel Functions

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29: 10.75 Tables

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§10.75(iii) Zeros and Associated Values of the Bessel Functions, Hankel Functions, and their Derivatives

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Döring (1966) tabulates all zeros of $Y_0\left(z\right)$, $Y_1\left(z\right)$, $H_0^{(1)}\left(z\right)$, $H_1^{(1)}\left(z\right)$, that lie in the sector , $|\mathrm{ph}z|\le \pi$, to 10D. Some of the smaller zeros of $Y_n\left(z\right)$ and $H_n^{(1)}\left(z\right)$ for $n=2,3,4,5,15$ are also included.

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Kerimov and Skorokhodov (1985b) tabulates 50 zeros of the principal branches of $H_0^{(1)}\left(z\right)$ and $H_1^{(1)}\left(z\right)$, 8D.

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Kerimov and Skorokhodov (1987) tabulates 100 complex double zeros $\nu$ of $Y_{\nu }^{\prime }\left(z{\mathrm{e}}^{-\pi \mathrm{i}}\right)$ and $H_{\nu }^{(1)}{}_{}{}^{\prime }\left(z{\mathrm{e}}^{-\pi \mathrm{i}}\right)$, 8D.

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§10.75(iv) Integrals of Bessel Functions

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30: 10.20 Uniform Asymptotic Expansions for Large Order

§10.20 Uniform Asymptotic Expansions for Large Order

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10.20.6 $$\begin{array}{c}{H}_{\nu }^{(1)}\left(\nu z\right)\\ H_{\nu }^{(2)}\left(\nu z\right)\end{array}\}\sim 2{\mathrm{e}}^{\mp \pi \mathrm{i}/3}{\left(\frac{4\zeta }{1-z^2}\right)}^{\frac{1}{4}}\left(\frac{\mathrm{Ai}\left({\mathrm{e}}^{\pm 2\pi \mathrm{i}/3}{\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{1}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{A_k(\zeta )}{{\nu }^{2k}}+\frac{{\mathrm{e}}^{\pm 2\pi \mathrm{i}/3}{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{\pm 2\pi \mathrm{i}/3}{\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{5}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{B_k(\zeta )}{{\nu }^{2k}}\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\zeta (z)$: solution, $A_k(\zeta )$: coefficients and $B_k(\zeta )$: coefficients

A&S Ref:

9.3.37

Referenced by:

§10.20(iii), §10.41(v)

Permalink:

http://dlmf.nist.gov/10.20.E6

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See also:

Annotations for §10.20(i), §10.20 and Ch.10

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10.20.9 $$\begin{array}{c}H_{\nu }^{(1)}{}_{}{}^{\prime }\left(\nu z\right)\\ H_{\nu }^{(2)}{}_{}{}^{\prime }\left(\nu z\right)\end{array}\}\sim \frac{4{\mathrm{e}}^{\mp 2\pi \mathrm{i}/3}}{z}{\left(\frac{1-z^2}{4\zeta }\right)}^{\frac{1}{4}}\left(\frac{{\mathrm{e}}^{\mp 2\pi \mathrm{i}/3}\mathrm{Ai}\left({\mathrm{e}}^{\pm 2\pi \mathrm{i}/3}{\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{4}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{C_k(\zeta )}{{\nu }^{2k}}+\frac{{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{\pm 2\pi \mathrm{i}/3}{\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{2}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{D_k(\zeta )}{{\nu }^{2k}}\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\zeta (z)$: solution, $C_k(\zeta )$: coefficients and $D_k(\zeta )$: coefficients

A&S Ref:

9.3.45

Referenced by:

§10.20(ii)

Permalink:

http://dlmf.nist.gov/10.20.E9

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See also:

Annotations for §10.20(i), §10.20 and Ch.10

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§10.20(iii) Double Asymptotic Properties

►For asymptotic properties of the expansions (10.20.4)–(10.20.6) with respect to large values of $z$ see §10.41(v).