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31: 10.40 Asymptotic Expansions for Large Argument

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§10.40(i) Hankel’s Expansions

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32: 10.61 Definitions and Basic Properties

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10.61.2 $${\ker }_{\nu }x+\mathrm{i}{\mathrm{kei}}_{\nu }x={\mathrm{e}}^{-\nu \pi \mathrm{i}/2}{K}_{\nu }\left(x{\mathrm{e}}^{\pi \mathrm{i}/4}\right)=\frac{1}{2}\pi \mathrm{i}{H}_{\nu }^{(1)}\left(x{\mathrm{e}}^{3\pi \mathrm{i}/4}\right)=-\frac{1}{2}\pi \mathrm{i}{\mathrm{e}}^{-\nu \pi \mathrm{i}}{H}_{\nu }^{(2)}\left(x{\mathrm{e}}^{-\pi \mathrm{i}/4}\right).$$

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Defines:

${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function and ${\ker }_{\nu }\left(x\right)$: Kelvin function

Symbols:

$H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.9.2

Referenced by:

§10.63(ii), §10.65(ii), §10.67(i), §10.69

Permalink:

http://dlmf.nist.gov/10.61.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.61(i), §10.61 and Ch.10

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33: 28.23 Expansions in Series of Bessel Functions

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${𝒞}_{\mu }^{(3)}=H_{\mu }^{(1)},$

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${𝒞}_{\mu }^{(4)}=H_{\mu }^{(2)};$

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34: 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). …

35: 10.41 Asymptotic Expansions for Large Order

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§10.41(v) Double Asymptotic Properties (Continued)

… ►We first prove that for the expansions (10.20.6) for the Hankel functions $H_{\nu }^{(1)}\left(\nu z\right)$ and $H_{\nu }^{(2)}\left(\nu z\right)$ the $z$-asymptotic property applies when $z\to \pm \mathrm{i}\mathrm{\infty }$, respectively. …We then extend the validity of this property from $z\to \pm \mathrm{i}\mathrm{\infty }$ to $z\to \mathrm{\infty }$ in the sector $-\pi +\delta \le \mathrm{ph}z\le 2\pi -\delta$ in the case of $H_{\nu }^{(1)}\left(\nu z\right)$, and to $z\to \mathrm{\infty }$ in the sector $-2\pi +\delta \le \mathrm{ph}z\le \pi -\delta$ in the case of $H_{\nu }^{(2)}\left(\nu z\right)$. …

36: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function

… ►For incomplete modified Bessel functions and Hankel functions, including applications, see Cicchetti and Faraone (2004).

37: 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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38: 30.11 Radial Spheroidal Wave Functions

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${𝒞}_{\nu }^{(3)}=H_{\nu }^{(1)},$

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${𝒞}_{\nu }^{(4)}=H_{\nu }^{(2)},$

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

39: 10.21 Zeros

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Zeros of $H_n^{(1)}\left(nz\right)$, $H_n^{(2)}\left(nz\right)$, $H_n^{(1)}{}_{}{}^{\prime }\left(nz\right)$, $H_n^{(2)}{}_{}{}^{\prime }\left(nz\right)$

… ►The first set of zeros of the principal value of $H_n^{(1)}\left(nz\right)$ is an infinite string with asymptote $\mathrm{\Im }z=-\mathrm{i}d/n$, where … ►The zeros of $H_n^{(1)}{}_{}{}^{\prime }\left(nz\right)$ have a similar pattern to those of $H_n^{(1)}\left(nz\right)$. The zeros of $H_n^{(2)}\left(nz\right)$ and $H_n^{(2)}{}_{}{}^{\prime }\left(nz\right)$ are the complex conjugates of the zeros of $H_n^{(1)}\left(nz\right)$ and $H_n^{(1)}{}_{}{}^{\prime }\left(nz\right)$, respectively. … ► …

40: 10.22 Integrals

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Products

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Trigonometric Arguments

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Convolutions

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§10.22(v) Hankel Transform

… ►Hankel’s inversion theorem is given by …