Bessel functions and Hankel functions

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21: 10.6 Recurrence Relations and Derivatives

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§10.6(i) Recurrence Relations

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§10.6(ii) Derivatives

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22: 10.9 Integral Representations

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Mehler–Sonine and Related Integrals

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Schläfli–Sommerfeld Integrals

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$H_{\nu }^{(2)}\left(z\right)=-{\displaystyle \frac{1}{\pi \mathrm{i}}}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }-\pi \mathrm{i}}}{\mathrm{e}}^{z\sinh t-\nu t}dt.$

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§10.9(iv) Compendia

►For collections of integral representations of Bessel and Hankel functions see Erdélyi et al. (1953b, §§7.3 and 7.12), Erdélyi et al. (1954a, pp. 43–48, 51–60, 99–105, 108–115, 123–124, 272–276, and 356–357), Gröbner and Hofreiter (1950, pp. 189–192), Marichev (1983, pp. 191–192 and 196–210), Magnus et al. (1966, §3.6), and Watson (1944, Chapter 6).

23: 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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24: 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. … ►

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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10.47.15 ${𝗁}_n^{(1)}\left(-z\right)={(-1)}^n{𝗁}_n^{(2)}\left(z\right),$ ${𝗁}_n^{(2)}\left(-z\right)={(-1)}^n{𝗁}_n^{(1)}\left(z\right).$

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Symbols:

${𝗁}_n^{(1)}\left(z\right)$: 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

Permalink:

http://dlmf.nist.gov/10.47.E15

Encodings:

pMML, pMML, png, png, TeX, TeX

See also:

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

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25: 10.8 Power Series

§10.8 Power Series

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26: 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. …

27: 9.17 Methods of Computation

… ►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. …

28: 10.52 Limiting Forms

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10.52.2 $$-{𝗒}_n\left(z\right),\mathrm{i}{𝗁}_n^{(1)}\left(z\right),-\mathrm{i}{𝗁}_n^{(2)}\left(z\right),{(-1)}^n{𝗂}_n^{(2)}\left(z\right),(2/\pi ){𝗄}_n\left(z\right)\sim (2n-1)!!/z^{n+1}.$$

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Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $!!$: double factorial, $\mathrm{i}$: imaginary unit, ${𝗂}_n^{(2)}\left(z\right)$: modified spherical Bessel function, ${𝗄}_n\left(z\right)$: modified spherical Bessel function, ${𝗒}_n\left(z\right)$: spherical Bessel function of the second kind, ${𝗁}_n^{(1)}\left(z\right)$: 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

A&S Ref:

10.1.5

Permalink:

http://dlmf.nist.gov/10.52.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.52(i), §10.52 and Ch.10

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29: 10.22 Integrals

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Products

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Trigonometric Arguments

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Orthogonality

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Orthogonality

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§10.22(v) Hankel Transform

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30: 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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