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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 ${\mathsf{j}}_n\left(z\right)$, ${\mathsf{y}}_n\left(z\right)$, ${\mathsf{h}}_n^{(1)}\left(z\right)$, ${\mathsf{h}}_n^{(2)}\left(z\right)$; modified spherical Bessel functions ${\mathsf{i}}_n^{(1)}\left(z\right)$, ${\mathsf{i}}_n^{(2)}\left(z\right)$, ${\mathsf{k}}_n\left(z\right)$; Kelvin functions ${\mathrm{ber}}_{\nu }\left(x\right)$, ${\mathrm{bei}}_{\nu }\left(x\right)$, ${\ker }_{\nu }\left(x\right)$, ${\mathrm{kei}}_{\nu }\left(x\right)$. … ►Abramowitz and Stegun (1964): $j_n(z)$, $y_n(z)$, $h_n^{(1)}(z)$, $h_n^{(2)}(z)$, for ${\mathsf{j}}_n\left(z\right)$, ${\mathsf{y}}_n\left(z\right)$, ${\mathsf{h}}_n^{(1)}\left(z\right)$, ${\mathsf{h}}_n^{(2)}\left(z\right)$, respectively, when $n\ge 0$. ►Jeffreys and Jeffreys (1956): ${\mathrm{Hs}}_{\nu }(z)$ for $H_{\nu }^{(1)}\left(z\right)$, ${\mathrm{Hi}}_{\nu }(z)$ for $H_{\nu }^{(2)}\left(z\right)$, ${\mathrm{Kh}}_{\nu }(z)$ for $(2/\pi )K_{\nu }\left(z\right)$. … ►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).

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

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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), $𝒲$: 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

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

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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), $𝒲$: 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

3: 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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10.11.7 $$H_n^{(1)}\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={(-1)}^{mn-1}((m-1)H_n^{(1)}\left(z\right)+m{H}_n^{(2)}\left(z\right)),$$

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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

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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

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$H_{\nu }^{(1)}\left(\overline{z}\right)=\overline{H_{\nu }^{(2)}\left(z\right)},$ $H_{\nu }^{(2)}\left(\overline{z}\right)=\overline{H_{\nu }^{(1)}\left(z\right)}.$

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4: 10.4 Connection Formulas

§10.4 Connection Formulas

►Other solutions of (10.2.1) include $J_{-\nu }\left(z\right)$, $Y_{-\nu }\left(z\right)$, $H_{-\nu }^{(1)}\left(z\right)$, and $H_{-\nu }^{(2)}\left(z\right)$. … ►

$$H_{-n}^{(1)}\left(z\right)={(-1)}^n{H}_n^{(1)}\left(z\right),$$

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$$H_{-n}^{(2)}\left(z\right)={(-1)}^n{H}_n^{(2)}\left(z\right).$$

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$$J_{\nu }\left(z\right)=\frac{1}{2}\left(H_{\nu }^{(1)}\left(z\right)+H_{\nu }^{(2)}\left(z\right)\right),$$

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5: 10.2 Definitions

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Bessel Functions of the Third Kind (Hankel Functions)

►These solutions of (10.2.1) are denoted by $H_{\nu }^{(1)}\left(z\right)$ and $H_{\nu }^{(2)}\left(z\right)$, and their defining properties are given by … ►The principal branches of $H_{\nu }^{(1)}\left(z\right)$ and $H_{\nu }^{(2)}\left(z\right)$ are two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi$. … ►

Branch Conventions

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6: 10.47 Definitions and Basic Properties

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

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Defines:

${\mathsf{h}}_n^{(1)}\left(z\right)$: 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

… ► ${\mathsf{j}}_n\left(z\right)$ and ${\mathsf{y}}_n\left(z\right)$ are the spherical Bessel functions of the first and second kinds, respectively; ${\mathsf{h}}_n^{(1)}\left(z\right)$ and ${\mathsf{h}}_n^{(2)}\left(z\right)$ are the spherical Bessel functions of the third kind. … ►Many properties of ${\mathsf{j}}_n\left(z\right)$, ${\mathsf{y}}_n\left(z\right)$, ${\mathsf{h}}_n^{(1)}\left(z\right)$, ${\mathsf{h}}_n^{(2)}\left(z\right)$, ${\mathsf{i}}_n^{(1)}\left(z\right)$, ${\mathsf{i}}_n^{(2)}\left(z\right)$, and ${\mathsf{k}}_n\left(z\right)$ follow straightforwardly from the above definitions and results given in preceding sections of this chapter. For example, $z^{-n}{\mathsf{j}}_n\left(z\right)$, $z^{n+1}{\mathsf{y}}_n\left(z\right)$, $z^{n+1}{\mathsf{h}}_n^{(1)}\left(z\right)$, $z^{n+1}{\mathsf{h}}_n^{(2)}\left(z\right)$, $z^{-n}{\mathsf{i}}_n^{(1)}\left(z\right)$, $z^{n+1}{\mathsf{i}}_n^{(2)}\left(z\right)$, and $z^{n+1}{\mathsf{k}}_n\left(z\right)$ are all entire functions of $z$. … ►

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

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Symbols:

${\mathsf{h}}_n^{(1)}\left(z\right)$: spherical Bessel function of the third kind, ${\mathsf{h}}_n^{(2)}\left(z\right)$: 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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7: 10.16 Relations to Other Functions

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

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

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Confluent Hypergeometric Functions

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

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

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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

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8: 9.6 Relations to Other Functions

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

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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

Source:

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

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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

Source:

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

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9.6.8 $$\mathrm{Bi}\left(-z\right)=\sqrt{z/3}\left(J_{-1/3}\left(\zeta \right)-J_{1/3}\left(\zeta \right)\right)=\frac{1}{2}\sqrt{z/3}\left({\mathrm{e}}^{2\pi \mathrm{i}/3}{H}_{1/3}^{(1)}\left(\zeta \right)+{\mathrm{e}}^{-2\pi \mathrm{i}/3}{H}_{1/3}^{(2)}\left(\zeta \right)\right)=\frac{1}{2}\sqrt{z/3}\left({\mathrm{e}}^{\pi \mathrm{i}/3}{H}_{-1/3}^{(1)}\left(\zeta \right)+{\mathrm{e}}^{-\pi \mathrm{i}/3}{H}_{-1/3}^{(2)}\left(\zeta \right)\right),$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: 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

Source:

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

A&S Ref:

10.4.19 (in slightly different form)

Permalink:

http://dlmf.nist.gov/9.6.E8

Encodings:

pMML, png, TeX

See also:

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

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

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9: 10.7 Limiting Forms

§10.7 Limiting Forms

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10.7.2 $$H_0^{(1)}\left(z\right)\sim -H_0^{(2)}\left(z\right)\sim (2\mathrm{i}/\pi )\ln z,$$

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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), $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

9.1.8

Referenced by:

§10.7(i)

Permalink:

http://dlmf.nist.gov/10.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.7(i), §10.7 and Ch.10

… ►For $H_{-\nu }^{(1)}\left(z\right)$ and $H_{-\nu }^{(2)}\left(z\right)$ when $\mathrm{\Re }\nu >0$ combine (10.4.6) and (10.7.7). For $H_{\mathrm{i}\nu }^{(1)}\left(z\right)$ and $H_{\mathrm{i}\nu }^{(2)}\left(z\right)$ when $\nu \in ℝ$ and $\nu \ne 0$ combine (10.4.3), (10.7.3), and (10.7.6). … ►For the corresponding results for $H_{\nu }^{(1)}\left(z\right)$ and $H_{\nu }^{(2)}\left(z\right)$ see (10.2.5) and (10.2.6).

10: 10.3 Graphics

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§10.3(ii) Real Order, Complex Variable

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Figure 10.3.10: $H_0^{(1)}\left(x+\mathrm{i}y\right)$, $-10\le x\le 5$, $-2.8\le y\le 4$. … Magnify 3D Help

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Figure 10.3.12: $H_1^{(1)}\left(x+\mathrm{i}y\right)$, $-10\le x\le 5$, $-2.8\le y\le 4$. … Magnify 3D Help

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Figure 10.3.14: $H_5^{(1)}\left(x+\mathrm{i}y\right)$, $-20\le x\le 10$, $-4\le y\le 4$. … Magnify 3D Help

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Figure 10.3.16: $H_{5.5}^{(1)}\left(x+\mathrm{i}y\right)$, $-20\le x\le 10$, $-4\le y\le 4$. … Magnify 3D Help

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