Hankel functions

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1: 10.1 Special Notation

… ►The main functions treated in this chapter are the Bessel functions $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$; Hankel functions $H_{\nu }^{(1)}\left(z\right)$, $H_{\nu }^{(2)}\left(z\right)$; modified Bessel functions $I_{\nu }\left(z\right)$, $K_{\nu }\left(z\right)$; spherical Bessel functions ${𝗃}_n\left(z\right)$, ${𝗒}_n\left(z\right)$, ${𝗁}_n^{(1)}\left(z\right)$, ${𝗁}_n^{(2)}\left(z\right)$; modified spherical Bessel functions ${𝗂}_n^{(1)}\left(z\right)$, ${𝗂}_n^{(2)}\left(z\right)$, ${𝗄}_n\left(z\right)$; Kelvin functions ${\mathrm{ber}}_{\nu }\left(x\right)$, ${\mathrm{bei}}_{\nu }\left(x\right)$, ${\ker }_{\nu }\left(x\right)$, ${\mathrm{kei}}_{\nu }\left(x\right)$. … ►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).

2: 10.76 Approximations

… ►

§10.76(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions

…

3: 10.4 Connection Formulas

§10.4 Connection Formulas

… ►

10.4.7 $$H_{\nu }^{(1)}\left(z\right)=\mathrm{i}\csc \left(\nu \pi \right)\left({\mathrm{e}}^{-\nu \pi \mathrm{i}}{J}_{\nu }\left(z\right)-J_{-\nu }\left(z\right)\right)=\csc \left(\nu \pi \right)\left(Y_{-\nu }\left(z\right)-{\mathrm{e}}^{-\nu \pi \mathrm{i}}{Y}_{\nu }\left(z\right)\right),$$

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Symbols:

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

Referenced by:

§10.4, §10.8

Permalink:

http://dlmf.nist.gov/10.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.4 and Ch.10

►

10.4.8 $$H_{\nu }^{(2)}\left(z\right)=\mathrm{i}\csc \left(\nu \pi \right)\left(J_{-\nu }\left(z\right)-{\mathrm{e}}^{\nu \pi \mathrm{i}}{J}_{\nu }\left(z\right)\right)=\csc \left(\nu \pi \right)\left(Y_{-\nu }\left(z\right)-{\mathrm{e}}^{\nu \pi \mathrm{i}}{Y}_{\nu }\left(z\right)\right).$$

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Symbols:

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

Referenced by:

§10.4, §10.8

Permalink:

http://dlmf.nist.gov/10.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.4 and Ch.10

…

4: 10.5 Wronskians and Cross-Products

§10.5 Wronskians and Cross-Products

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10.5.3 $$𝒲\left\{J_{\nu }\left(z\right),H_{\nu }^{(1)}\left(z\right)\right\}=J_{\nu +1}\left(z\right)H_{\nu }^{(1)}\left(z\right)-J_{\nu }\left(z\right)H_{\nu +1}^{(1)}\left(z\right)=2\mathrm{i}/(\pi z),$$

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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), $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.5 and Ch.10

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10.5.4 $$𝒲\left\{J_{\nu }\left(z\right),H_{\nu }^{(2)}\left(z\right)\right\}=J_{\nu +1}\left(z\right)H_{\nu }^{(2)}\left(z\right)-J_{\nu }\left(z\right)H_{\nu +1}^{(2)}\left(z\right)=-2\mathrm{i}/(\pi z),$$

ⓘ

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), $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.5 and Ch.10

►

10.5.5 $$𝒲\left\{H_{\nu }^{(1)}\left(z\right),H_{\nu }^{(2)}\left(z\right)\right\}=H_{\nu +1}^{(1)}\left(z\right)H_{\nu }^{(2)}\left(z\right)-H_{\nu }^{(1)}\left(z\right)H_{\nu +1}^{(2)}\left(z\right)=-4\mathrm{i}/(\pi z).$$

ⓘ

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), $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.17

Permalink:

http://dlmf.nist.gov/10.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.5 and Ch.10

5: 10.11 Analytic Continuation

§10.11 Analytic Continuation

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10.11.3 $$\sin \left(\nu \pi \right)H_{\nu }^{(1)}\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)=-\sin \left((m-1)\nu \pi \right)H_{\nu }^{(1)}\left(z\right)-{\mathrm{e}}^{-\nu \pi \mathrm{i}}\sin \left(m\nu \pi \right)H_{\nu }^{(2)}\left(z\right),$$

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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), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $m$: integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.37

Referenced by:

§10.17(i)

Permalink:

http://dlmf.nist.gov/10.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.11 and Ch.10

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10.11.4 $$\sin \left(\nu \pi \right)H_{\nu }^{(2)}\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={\mathrm{e}}^{\nu \pi \mathrm{i}}\sin \left(m\nu \pi \right)H_{\nu }^{(1)}\left(z\right)+\sin \left((m+1)\nu \pi \right)H_{\nu }^{(2)}\left(z\right).$$

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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), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $m$: integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.38

Referenced by:

§10.11, §10.17(i)

Permalink:

http://dlmf.nist.gov/10.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.11 and Ch.10

… ►

10.11.7 $$H_n^{(1)}\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={(-1)}^{mn-1}((m-1)H_n^{(1)}\left(z\right)+m{H}_n^{(2)}\left(z\right)),$$

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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), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $m$: integer, $n$: integer and $z$: complex variable

Referenced by:

§10.17(i)

Permalink:

http://dlmf.nist.gov/10.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.11 and Ch.10

►

10.11.8 $$H_n^{(2)}\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={(-1)}^{mn}(m{H}_n^{(1)}\left(z\right)+(m+1)H_n^{(2)}\left(z\right)).$$

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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), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $m$: integer, $n$: integer and $z$: complex variable

Referenced by:

§10.11, §10.17(i)

Permalink:

http://dlmf.nist.gov/10.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.11 and Ch.10

…

6: 10.2 Definitions

… ►

§10.2(i) Bessel’s Equation

… ►

Bessel Functions of the Third Kind (Hankel Functions)

… ►

10.2.5 $$H_{\nu }^{(1)}\left(z\right)\sim \sqrt{2/(\pi z)}{\mathrm{e}}^{\mathrm{i}(z-\frac{1}{2}\nu \pi -\frac{1}{4}\pi )}$$

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Defines:

$H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function)

Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.2.3

Referenced by:

§10.2(ii), §10.7(ii)

Permalink:

http://dlmf.nist.gov/10.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.2(ii), §10.2(ii), §10.2 and Ch.10

… ► … ►

Branch Conventions

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7: 10.16 Relations to Other Functions

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$$H_{\frac{1}{2}}^{(2)}\left(z\right)=\mathrm{i}{H}_{-\frac{1}{2}}^{(2)}\left(z\right)=\mathrm{i}{\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}{\mathrm{e}}^{-\mathrm{i}z}.$$

… ►

Confluent Hypergeometric Functions

… ►

10.16.6 $$\begin{array}{c}{H}_{\nu }^{(1)}\left(z\right)\\ H_{\nu }^{(2)}\left(z\right)\end{array}\}=\mp 2{\pi }^{-\frac{1}{2}}\mathrm{i}{\mathrm{e}}^{\mp \nu \pi \mathrm{i}}{(2z)}^{\nu }{\mathrm{e}}^{\pm \mathrm{i}z}U(\nu +\frac{1}{2},2\nu +1,\mp 2\mathrm{i}z).$$

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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), $U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.16, §10.17(v)

Permalink:

http://dlmf.nist.gov/10.16.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.16, §10.16 and Ch.10

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10.16.8 $$\begin{array}{c}{H}_{\nu }^{(1)}\left(z\right)\\ H_{\nu }^{(2)}\left(z\right)\end{array}\}={\mathrm{e}}^{\mp (2\nu +1)\pi \mathrm{i}/4}{\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}{W}_{0,\nu }\left(\mp 2\mathrm{i}z\right).$$

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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), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.16, §10.16

Permalink:

http://dlmf.nist.gov/10.16.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.16, §10.16 and Ch.10

…

8: 9.6 Relations to Other Functions

… ►

§9.6(i) Airy Functions as Bessel Functions, Hankel Functions, and Modified Bessel Functions

… ►

9.6.6 $$\mathrm{Ai}\left(-z\right)=(\sqrt{z}/3)\left(J_{1/3}\left(\zeta \right)+J_{-1/3}\left(\zeta \right)\right)=\frac{1}{2}\sqrt{z/3}\left({\mathrm{e}}^{\pi \mathrm{i}/6}{H}_{1/3}^{(1)}\left(\zeta \right)+{\mathrm{e}}^{-\pi \mathrm{i}/6}{H}_{1/3}^{(2)}\left(\zeta \right)\right)=\frac{1}{2}\sqrt{z/3}\left({\mathrm{e}}^{-\pi \mathrm{i}/6}{H}_{-1/3}^{(1)}\left(\zeta \right)+{\mathrm{e}}^{\pi \mathrm{i}/6}{H}_{-1/3}^{(2)}\left(\zeta \right)\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $J_{\nu }\left(z\right)$: Bessel function of the first kind, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: 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, $z$: complex variable and $\zeta$: change of variable

Proof sketch:

Derivable from those given in Miller (1946, p. B17) and Olver (1997b, pp. 392–393).

A&S Ref:

10.4.15 (partial)

Permalink:

http://dlmf.nist.gov/9.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §9.6(i), §9.6 and Ch.9

►

9.6.7 $${\mathrm{Ai}}^{\prime }\left(-z\right)=(z/3)\left(J_{2/3}\left(\zeta \right)-J_{-2/3}\left(\zeta \right)\right)=\frac{1}{2}(z/\sqrt{3})\left({\mathrm{e}}^{-\pi \mathrm{i}/6}{H}_{2/3}^{(1)}\left(\zeta \right)+{\mathrm{e}}^{\pi \mathrm{i}/6}{H}_{2/3}^{(2)}\left(\zeta \right)\right)=\frac{1}{2}(z/\sqrt{3})\left({\mathrm{e}}^{-5\pi \mathrm{i}/6}{H}_{-2/3}^{(1)}\left(\zeta \right)+{\mathrm{e}}^{5\pi \mathrm{i}/6}{H}_{-2/3}^{(2)}\left(\zeta \right)\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $J_{\nu }\left(z\right)$: Bessel function of the first kind, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: 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, $z$: complex variable and $\zeta$: change of variable

Proof sketch:

Derivable from those given in Miller (1946, p. B17) and Olver (1997b, pp. 392–393).

A&S Ref:

10.4.17 (partial)

Permalink:

http://dlmf.nist.gov/9.6.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §9.6(i), §9.6 and Ch.9

… ►

§9.6(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions as Airy Functions

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9.6.20 $$H_{2/3}^{(2)}\left(\zeta \right)={\mathrm{e}}^{2\pi \mathrm{i}/3}{H}_{-2/3}^{(2)}\left(\zeta \right)={\mathrm{e}}^{-\pi \mathrm{i}/6}(\sqrt{3}/z)\left({\mathrm{Ai}}^{\prime }\left(-z\right)+\mathrm{i}{\mathrm{Bi}}^{\prime }\left(-z\right)\right).$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy 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, $z$: complex variable and $\zeta (z)$: change of variable

Proof sketch:

Derivable from those given in Olver (1997b, pp. 392–393).

A&S Ref:

10.4.29

Referenced by:

(9.8.10), (9.8.12)

Permalink:

http://dlmf.nist.gov/9.6.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §9.6(ii), §9.6 and Ch.9

…

9: 15.14 Integrals

… ►Hankel transforms of hypergeometric functions are given in Oberhettinger (1972, §1.17) and Erdélyi et al. (1954b, §8.17). …

10: 10.74 Methods of Computation

… ►The power-series expansions given in §§10.2 and 10.8, together with the connection formulas of §10.4, can be used to compute the Bessel and Hankel functions when the argument $x$ or $z$ is sufficiently small in absolute value. … ►For $z\in ℂ$ the function $H_{\nu }^{(1)}\left(z\right)$, for example, can always be computed in a stable manner in the sector $0\le \mathrm{ph}z\le \pi$ by integrating along rays towards the origin. … ►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). … ► ►

§10.74(vi) Zeros and Associated Values

…