Bessel and Hankel functions

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21: 10.27 Connection Formulas

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10.27.7 $$I_{\nu }\left(z\right)=\frac{1}{2}{\mathrm{e}}^{\mp \nu \pi \mathrm{i}/2}\left(H_{\nu }^{(1)}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}/2}\right)+H_{\nu }^{(2)}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}/2}\right)\right),$$ $-\pi \le \pm \mathrm{ph}z\le \frac{1}{2}\pi$.

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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, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.27.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.27 and Ch.10

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10.27.8

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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, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.4

Referenced by:

§10.16, §10.26(ii), §10.27, §10.27, §10.32(ii), §10.39, §10.40(ii), §10.40(iii), §10.41(i), §10.42, §10.43(ii), §10.44(i), §10.44(ii), §10.44(iii), §10.47(iv), (9.7.16), §9.7(iii)

Permalink:

http://dlmf.nist.gov/10.27.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.27 and Ch.10

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22: 10.22 Integrals

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Products

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Trigonometric Arguments

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Convolutions

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Fractional Integral

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§10.22(v) Hankel Transform

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23: 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. …

24: 10.23 Sums

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§10.23(i) Multiplication Theorem

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§10.23(ii) Addition Theorems

… ► … ►For collections of sums of series involving Bessel or Hankel functions see Erdélyi et al. (1953b, §7.15), Gradshteyn and Ryzhik (2000, §§8.51–8.53), Hansen (1975), Luke (1969b, §9.4), Prudnikov et al. (1986b, pp. 651–691 and 697–700), and Wheelon (1968, pp. 48–51).

25: 11.2 Definitions

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11.2.13 $$w={𝐇}_{\nu }\left(z\right)+A{J}_{\nu }\left(z\right)+B{H}_{\nu }^{(1)}\left(z\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), ${𝐇}_{\nu }\left(z\right)$: Struve function, $z$: complex variable, $\nu$: real or complex order, $A$: arbitrary constant and $B$: arbitrary constant

Referenced by:

§11.2(iii)

Permalink:

http://dlmf.nist.gov/11.2.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §11.2(iii), §11.2 and Ch.11

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11.2.14 $$w={𝐇}_{\nu }\left(z\right)+A{J}_{\nu }\left(z\right)+B{H}_{\nu }^{(2)}\left(z\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), ${𝐇}_{\nu }\left(z\right)$: Struve function, $z$: complex variable, $\nu$: real or complex order, $A$: arbitrary constant and $B$: arbitrary constant

Referenced by:

§11.2(iii)

Permalink:

http://dlmf.nist.gov/11.2.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §11.2(iii), §11.2 and Ch.11

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11.2.15 $$w={𝐊}_{\nu }\left(z\right)+A{H}_{\nu }^{(1)}\left(z\right)+B{H}_{\nu }^{(2)}\left(z\right).$$

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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), ${𝐊}_{\nu }\left(z\right)$: Struve function, $z$: complex variable, $\nu$: real or complex order, $A$: arbitrary constant and $B$: arbitrary constant

Referenced by:

§11.2(iii), §11.2(iii)

Permalink:

http://dlmf.nist.gov/11.2.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §11.2(iii), §11.2 and Ch.11

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26: 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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27: 10.51 Recurrence Relations and Derivatives

… ►Let $f_n(z)$ denote any of ${𝗃}_n\left(z\right)$, ${𝗒}_n\left(z\right)$, ${𝗁}_n^{(1)}\left(z\right)$, or ${𝗁}_n^{(2)}\left(z\right)$. …

28: 28.23 Expansions in Series of Bessel Functions

§28.23 Expansions in Series of Bessel Functions

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${𝒞}_{\mu }^{(1)}=J_{\mu },$

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${𝒞}_{\mu }^{(3)}=H_{\mu }^{(1)},$

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${𝒞}_{\mu }^{(4)}=H_{\mu }^{(2)};$

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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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30: 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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