Hankel transform

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21: Bibliography R

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Finite-sum rules for Macdonald’s functions and Hankel’s symbols. Integral Transform. Spec. Funct. 10 (2), pp. 115–124.

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External Links:

ISSN 1065-2469, Document, MathReview, ZentralBlatt

Referenced by:

§10.60(iii).

Encodings:

BibTeX

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22: 10.9 Integral Representations

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10.9.24 $$H_{\nu }^{(1)}\left(z\right)=-\frac{{\mathrm{e}}^{-\frac{1}{2}\nu \pi \mathrm{i}}}{2{\pi }^2}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}\mathrm{\Gamma }\left(t\right)\mathrm{\Gamma }\left(t-\nu \right){(-\frac{1}{2}\mathrm{i}z)}^{\nu -2t}dt,$$ ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: complex parameter and $c$: constant

Keywords:

Mellin transform

Referenced by:

§10.9(ii)

Permalink:

http://dlmf.nist.gov/10.9.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §10.9(ii), §10.9(ii), §10.9 and Ch.10

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10.9.25 $$H_{\nu }^{(2)}\left(z\right)=\frac{{\mathrm{e}}^{\frac{1}{2}\nu \pi \mathrm{i}}}{2{\pi }^2}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}\mathrm{\Gamma }\left(t\right)\mathrm{\Gamma }\left(t-\nu \right){(\frac{1}{2}\mathrm{i}z)}^{\nu -2t}dt,$$ .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: complex parameter and $c$: constant

Keywords:

Mellin transform

Referenced by:

§10.9(ii), §10.9(ii)

Permalink:

http://dlmf.nist.gov/10.9.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §10.9(ii), §10.9(ii), §10.9 and Ch.10

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23: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function

… ►The Laplace transform of $\varphi (\rho ,\beta ;z)$ can be expressed in terms of the Mittag-Leffler function: … ►For incomplete modified Bessel functions and Hankel functions, including applications, see Cicchetti and Faraone (2004).

24: Bibliography N

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On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.

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External Links:

ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt

Referenced by:

§28.26(ii).

Encodings:

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On an asymptotic expansion of the Kontorovich-Lebedev transform. Applicable Anal. 39 (4), pp. 249–263.

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External Links:

ISSN 0003-6811, Document, MathReview (Aloknath Chakrabarti), ZentralBlatt

Referenced by:

§10.43(v).

Encodings:

BibTeX

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On an asymptotic expansion of the Kontorovich-Lebedev transform. Methods Appl. Anal. 3 (1), pp. 98–108.

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External Links:

ISSN 1073-2772, FullText, MathReview (C. M. Joshi), ZentralBlatt

Referenced by:

§10.43(v).

Encodings:

BibTeX

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On the large argument asymptotics of the Lommel function via Stieltjes transforms. Asymptot. Anal. 91 (3-4), pp. 265–281.

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External Links:

ISSN 0921-7134, Document, MathReview Entry, ZentralBlatt

Referenced by:

§11.6(iii), §11.6(iii), §11.9(iii), §11.9(iii).

Encodings:

BibTeX

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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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25: 9.17 Methods of Computation

… ►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 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. …

26: 18.2 General Orthogonal Polynomials

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§18.2(vii) Quadratic Transformations

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§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. …