Hankel functions

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31: 13.21 Uniform Asymptotic Approximations for Large $\kappa$

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13.21.3 $$W_{-\kappa ,\mu }\left(x{\mathrm{e}}^{-\pi \mathrm{i}}\right)=\frac{\pi \sqrt{x}}{\mathrm{\Gamma }\left(\kappa +\frac{1}{2}\right)}{\mathrm{e}}^{\mu \pi \mathrm{i}}\left(H_{2\mu }^{(1)}\left(2\sqrt{x\kappa }\right)+\mathrm{env}{Y}_{2\mu }\left(2\sqrt{x\kappa }\right)O\left({\kappa }^{-\frac{1}{2}}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{env}{Y}_{\nu }\left(x\right)$: envelope of Bessel function $Y_{\nu }\left(x\right)$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $x$: real variable

Permalink:

http://dlmf.nist.gov/13.21.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.21(i), §13.21 and Ch.13

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13.21.4 $$W_{-\kappa ,\mu }\left(x{\mathrm{e}}^{\pi \mathrm{i}}\right)=\frac{\pi \sqrt{x}}{\mathrm{\Gamma }\left(\kappa +\frac{1}{2}\right)}{\mathrm{e}}^{-\mu \pi \mathrm{i}}\left(H_{2\mu }^{(2)}\left(2\sqrt{x\kappa }\right)+\mathrm{env}{Y}_{2\mu }\left(2\sqrt{x\kappa }\right)O\left({\kappa }^{-\frac{1}{2}}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{env}{Y}_{\nu }\left(x\right)$: envelope of Bessel function $Y_{\nu }\left(x\right)$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $x$: real variable

Referenced by:

§13.21(i)

Permalink:

http://dlmf.nist.gov/13.21.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §13.21(i), §13.21 and Ch.13

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13.21.15 $$W_{-\kappa ,\mu }\left(x{\mathrm{e}}^{-\pi \mathrm{i}}\right)=c(\kappa ,\mu )\mathrm{\Psi }(\kappa ,\mu ,x)\left(H_{2\mu }^{(1)}\left(\sqrt{\zeta \kappa }\right)+\mathrm{env}{Y}_{2\mu }\left(\sqrt{\zeta \kappa }\right)O\left({\kappa }^{-1}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{env}{Y}_{\nu }\left(x\right)$: envelope of Bessel function $Y_{\nu }\left(x\right)$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $c(\kappa ,\mu )$: function and $\mathrm{\Psi }(\kappa ,\mu ,x)$: function

Permalink:

http://dlmf.nist.gov/13.21.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §13.21(ii), §13.21 and Ch.13

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13.21.16 $$W_{-\kappa ,\mu }\left(x{\mathrm{e}}^{\pi \mathrm{i}}\right)={\mathrm{e}}^{-2\mu \pi \mathrm{i}}c(\kappa ,\mu )\mathrm{\Psi }(\kappa ,\mu ,x)\left(H_{2\mu }^{(2)}\left(\sqrt{\zeta \kappa }\right)+\mathrm{env}{Y}_{2\mu }\left(\sqrt{\zeta \kappa }\right)O\left({\kappa }^{-1}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{env}{Y}_{\nu }\left(x\right)$: envelope of Bessel function $Y_{\nu }\left(x\right)$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $c(\kappa ,\mu )$: function and $\mathrm{\Psi }(\kappa ,\mu ,x)$: function

Permalink:

http://dlmf.nist.gov/13.21.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §13.21(ii), §13.21 and Ch.13

… ►For the functions $J_{2\mu }$, $Y_{2\mu }$, $H_{2\mu }^{(1)}$, and $H_{2\mu }^{(2)}$ see §10.2(ii), and for the $\mathrm{env}$ functions associated with $J_{2\mu }$ and $Y_{2\mu }$ see §2.8(iv). …

32: 10.18 Modulus and Phase Functions

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10.18.1 $$M_{\nu }\left(x\right){\mathrm{e}}^{\mathrm{i}{\theta }_{\nu }\left(x\right)}=H_{\nu }^{(1)}\left(x\right),$$

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Defines:

$M_{\nu }\left(x\right)$: modulus of Bessel functions

Symbols:

$H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${\theta }_{\nu }\left(x\right)$: phase of Bessel functions, $x$: real variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.18.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.18(i), §10.18 and Ch.10

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10.18.2 $$N_{\nu }\left(x\right){\mathrm{e}}^{\mathrm{i}{\varphi }_{\nu }\left(x\right)}=H_{\nu }^{(1)}{}_{}{}^{\prime }\left(x\right),$$

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Defines:

$N_{\nu }\left(x\right)$: modulus of derivatives of Bessel functions

Symbols:

$H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${\varphi }_{\nu }\left(x\right)$: phase of derivatives of Bessel functions, $x$: real variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.18.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.18(i), §10.18 and Ch.10

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33: 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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34: Bibliography H

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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.

Encodings:

BibTeX

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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).

Encodings:

BibTeX

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35: 10.47 Definitions and Basic Properties

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10.47.5 $${𝗁}_n^{(1)}\left(z\right)=\sqrt{\frac{1}{2}\pi /z}{H}_{n+\frac{1}{2}}^{(1)}\left(z\right)={(-1)}^{n+1}\mathrm{i}\sqrt{\frac{1}{2}\pi /z}{H}_{-n-\frac{1}{2}}^{(1)}\left(z\right),$$

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Defines:

${𝗁}_n^{(1)}\left(z\right)$: spherical Bessel function of the third kind

Symbols:

$H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $n$: integer and $z$: complex variable

Referenced by:

§10.51(i)

Permalink:

http://dlmf.nist.gov/10.47.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.47(ii), §10.47(ii), §10.47 and Ch.10

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10.47.6 $${𝗁}_n^{(2)}\left(z\right)=\sqrt{\frac{1}{2}\pi /z}{H}_{n+\frac{1}{2}}^{(2)}\left(z\right)={(-1)}^n\mathrm{i}\sqrt{\frac{1}{2}\pi /z}{H}_{-n-\frac{1}{2}}^{(2)}\left(z\right).$$

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Defines:

${𝗁}_n^{(2)}\left(z\right)$: spherical Bessel function of the third kind

Symbols:

$H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/10.47.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.47(ii), §10.47(ii), §10.47 and Ch.10

► ${𝗃}_n\left(z\right)$ and ${𝗒}_n\left(z\right)$ are the spherical Bessel functions of the first and second kinds, respectively; ${𝗁}_n^{(1)}\left(z\right)$ and ${𝗁}_n^{(2)}\left(z\right)$ are the spherical Bessel functions of the third kind. … ►For example, $z^{-n}{𝗃}_n\left(z\right)$, $z^{n+1}{𝗒}_n\left(z\right)$, $z^{n+1}{𝗁}_n^{(1)}\left(z\right)$, $z^{n+1}{𝗁}_n^{(2)}\left(z\right)$, $z^{-n}{𝗂}_n^{(1)}\left(z\right)$, $z^{n+1}{𝗂}_n^{(2)}\left(z\right)$, and $z^{n+1}{𝗄}_n\left(z\right)$ are all entire functions of $z$. … ►

10.47.13 $${𝗄}_n\left(z\right)=-\frac{1}{2}\pi {\mathrm{i}}^n{𝗁}_n^{(1)}\left(\mathrm{i}z\right)=-\frac{1}{2}\pi {\mathrm{i}}^{-n}{𝗁}_n^{(2)}\left(-\mathrm{i}z\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, ${𝗄}_n\left(z\right)$: modified spherical Bessel function, ${𝗁}_n^{(1)}\left(z\right)$: spherical Bessel function of the third kind, ${𝗁}_n^{(2)}\left(z\right)$: spherical Bessel function of the third kind, $n$: integer and $z$: complex variable

Referenced by:

§10.49(ii), §10.60(i)

Permalink:

http://dlmf.nist.gov/10.47.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §10.47(iv), §10.47 and Ch.10

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36: 10.21 Zeros

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§10.21(ix) Complex Zeros

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§10.21(xiv) $\nu$-Zeros

►For information on zeros of Bessel and Hankel functions as functions of the order, see Cochran (1965), Cochran and Hoffspiegel (1970), Hethcote (1970), Conde and Kalla (1979), and Sandström and Ackrén (2007).

37: 30.11 Radial Spheroidal Wave Functions

§30.11 Radial Spheroidal Wave Functions

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§30.11(i) Definitions

… ►with $J_{\nu }$, $Y_{\nu }$, $H_{\nu }^{(1)}$, and $H_{\nu }^{(2)}$ as in §10.2(ii). … ►

Connection Formulas

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§30.11(ii) Graphics

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38: 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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39: 13.10 Integrals

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§13.10(v) Hankel Transforms

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40: Software Index

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	Open Source													With Book						Commercial
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10.77(v) Bessel Functions–Real Order and Complex Argument (including Hankel Functions)	✓	✓									✓	a	✓						✓	✓	✓	✓	✓
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