Hankel integrals

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11: Bibliography R

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Computation of Hankel (Bessel) functions of complex index and argument by numerical integration of a Schläfli contour integral. Ž. Vyčisl. Mat. i Mat. Fiz. 13, pp. 1415–1424, 1636.

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Notes:

English translation in U.S.S.R. Computational Math. and Math. Phys. 13(6), pp. 58–67

External Links:

ISSN 0044-4669, Document, MathReview (H. Zegeling), ZentralBlatt

Referenced by:

§10.74(iii).

Encodings:

BibTeX

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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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12: 13.4 Integral Representations

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13.4.2 $$𝐌(a,b,z)=\frac{1}{\mathrm{\Gamma }\left(b-c\right)}{\int }_0^1𝐌(a,c,zt)t^{c-1}{(1-t)}^{b-c-1}dt,$$ $\mathrm{\Re }b>\mathrm{\Re }c>0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable and $c$: arbitrary

Referenced by:

§13.10(vi)

Permalink:

http://dlmf.nist.gov/13.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(i), §13.4 and Ch.13

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13: 25.11 Hurwitz Zeta Function

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25.11.30 $$\zeta (s,a)=\frac{\mathrm{\Gamma }\left(1-s\right)}{2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+)}\frac{{\mathrm{e}}^{az}{z}^{s-1}}{1-{\mathrm{e}}^z}dz,$$ $s\ne 1$, $\mathrm{\Re }a>0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta 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, $\mathrm{\Re }$: real part, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable

Keywords:

contour integral, integral representation

Source:

Erdélyi et al. (1953a, (1.10.5), p. 25); with $z=-t$

Proof sketch:

This contour integral uses Hankel’s contour Figure 5.9.1. Assume $\mathrm{\Re }s>1$, collapse the integration path onto the negative real axis, apply (25.11.25), followed by analytic continuation for all $s\ne 1$, $\mathrm{\Re }a>0$, since the integrand is analytic in $z$, the convergence at the ends of the path is exponential for all $s\ne -1$ and the Hankel contour can be kept well clear of the singularity at the origin in the $z$-plane.

Referenced by:

§25.11(i)

Permalink:

http://dlmf.nist.gov/25.11.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(vii), §25.11 and Ch.25

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14: 10.74 Methods of Computation

… ►For evaluation of the Hankel functions $H_{\nu }^{(1)}\left(z\right)$ and $H_{\nu }^{(2)}\left(z\right)$ for complex values of $\nu$ and $z$ based on the integral representations (10.9.18) see Remenets (1973). …

15: Bibliography L

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Integral representation of the Hankel function in terms of parabolic cylinder functions. Quart. J. Mech. Appl. Math. 23 (3), pp. 315–327.

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External Links:

Document, MathReview (T. Erber), ZentralBlatt

Referenced by:

§12.12, §12.16.

Encodings:

BibTeX

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16: 10.54 Integral Representations

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${𝗁}_n^{(1)}\left(z\right)={\displaystyle \frac{{(-\mathrm{i})}^{n+1}}{\pi }}{\displaystyle {\int }_{\mathrm{i}\mathrm{\infty }}^{(1+)}}{\mathrm{e}}^{\mathrm{i}zt}{Q}_n\left(t\right)dt,$

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${𝗁}_n^{(2)}\left(z\right)={\displaystyle \frac{{(-\mathrm{i})}^{n+1}}{\pi }}{\displaystyle {\int }_{\mathrm{i}\mathrm{\infty }}^{(-1+)}}{\mathrm{e}}^{\mathrm{i}zt}{Q}_n\left(t\right)dt,$

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17: 10.43 Integrals

§10.43 Integrals

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§10.43(i) Indefinite Integrals

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§10.43(iii) Fractional Integrals

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18: 3.5 Quadrature

… ►Other contour integrals occur in standard integral transforms or their inverses, for example, Hankel transforms (§10.22(v)), Kontorovich–Lebedev transforms (§10.43(v)), and Mellin transforms (§1.14(iv)). …

19: 1.14 Integral Transforms

§1.14 Integral Transforms

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20: Software Index

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	Open Source													With Book						Commercial
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6 Exponential, Logarithmic, Sine, and Cosine Integrals
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10.77(v) Bessel Functions–Real Order and Complex Argument (including Hankel Functions)	✓	✓									✓	a	✓						✓	✓	✓	✓	✓
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11.16(iii) Integrals of …														✓					✓		✓	✓	a
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11.16(vi) Integrals of …														✓							✓	✓	a
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19 Elliptic Integrals
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