10 Bessel FunctionsBessel and Hankel Functions10.10 Continued Fractions10.12 Generating Function and Associated Series

§10.11 Analytic Continuation

ⓘ

Keywords:

Bessel functions, Hankel functions, analytic continuation

Notes:

For (10.11.1)–(10.11.5) use (10.2.2), (10.2.3), (10.4.3). For (10.11.6)–(10.11.8) take limits. For (10.11.9) use the Schwarz Reflection Principle (§1.10(ii)).

Referenced by:

§10.4, Erratum (V1.0.19) for Notation

Permalink:

http://dlmf.nist.gov/10.11

See also:

Annotations for Ch.10

When $m\in \mathbb{Z}$,

10.11.1		$$J_{\nu }\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={\mathrm{e}}^{m\nu \pi \mathrm{i}}{J}_{\nu }\left(z\right),$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $m$: integer, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.1.35 Referenced by: §10.11, §10.17(i), §10.47(v) Permalink: http://dlmf.nist.gov/10.11.E1 Encodings: TeX, pMML, png See also: Annotations for §10.11 and Ch.10

10.11.2		$$Y_{\nu }\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={\mathrm{e}}^{-m\nu \pi \mathrm{i}}{Y}_{\nu }\left(z\right)+2\mathrm{i}\sin \left(m\nu \pi \right)\cot \left(\nu \pi \right)J_{\nu }\left(z\right).$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $m$: integer, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.1.36 Referenced by: §10.11, §10.47(v) Permalink: http://dlmf.nist.gov/10.11.E2 Encodings: TeX, pMML, png See also: Annotations for §10.11 and Ch.10

10.11.3		$$\begin{array}{l}\sin \left(\nu \pi \right)H_{\nu }^{(1)}\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)\\ =-\sin \left((m-1)\nu \pi \right)H_{\nu }^{(1)}\left(z\right)-{\mathrm{e}}^{-\nu \pi \mathrm{i}}\sin \left(m\nu \pi \right)H_{\nu }^{(2)}\left(z\right),\end{array}$$
ⓘ 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: TeX, pMML, png See also: Annotations for §10.11 and Ch.10

10.11.4		$$\sin \left(\nu \pi \right)H_{\nu }^{(2)}\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={\mathrm{e}}^{\nu \pi \mathrm{i}}\sin \left(m\nu \pi \right)H_{\nu }^{(1)}\left(z\right)+\sin \left((m+1)\nu \pi \right)H_{\nu }^{(2)}\left(z\right).$$
ⓘ 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: TeX, pMML, png See also: Annotations for §10.11 and Ch.10

10.11.5		$H_{\nu }^{(1)}\left(z{\mathrm{e}}^{\pi \mathrm{i}}\right)$	$=-{\mathrm{e}}^{-\nu \pi \mathrm{i}}{H}_{\nu }^{(2)}\left(z\right),$
		$H_{\nu }^{(2)}\left(z{\mathrm{e}}^{-\pi \mathrm{i}}\right)$	$=-{\mathrm{e}}^{\nu \pi \mathrm{i}}{H}_{\nu }^{(1)}\left(z\right).$
ⓘ 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, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.1.39 Referenced by: §10.11, §10.27 Permalink: http://dlmf.nist.gov/10.11.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.11 and Ch.10

If $\nu =n$ $(\in \mathbb{Z})$, then limiting values are taken in (10.11.2)–(10.11.4):

10.11.6		$$Y_n\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={(-1)}^{mn}(Y_n\left(z\right)+2\mathrm{i}m{J}_n\left(z\right)),$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: 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 Permalink: http://dlmf.nist.gov/10.11.E6 Encodings: TeX, pMML, png See also: Annotations for §10.11 and Ch.10

10.11.7		$$H_n^{(1)}\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={(-1)}^{mn-1}((m-1)H_n^{(1)}\left(z\right)+m{H}_n^{(2)}\left(z\right)),$$
ⓘ 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: TeX, pMML, png See also: Annotations for §10.11 and Ch.10

10.11.8		$$H_n^{(2)}\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={(-1)}^{mn}(m{H}_n^{(1)}\left(z\right)+(m+1)H_n^{(2)}\left(z\right)).$$
ⓘ 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: TeX, pMML, png See also: Annotations for §10.11 and Ch.10

For real $\nu$,

10.11.9		$J_{\nu }\left(\overline{z}\right)$	$=\overline{J_{\nu }\left(z\right)},$	$Y_{\nu }\left(\overline{z}\right)$	$=\overline{Y_{\nu }\left(z\right)},$
		$H_{\nu }^{(1)}\left(\overline{z}\right)$	$=\overline{H_{\nu }^{(2)}\left(z\right)},$	$H_{\nu }^{(2)}\left(\overline{z}\right)$	$=\overline{H_{\nu }^{(1)}\left(z\right)}.$
ⓘ 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), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\overline{z}$: complex conjugate, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.1.40 Referenced by: §10.11 Permalink: http://dlmf.nist.gov/10.11.E9 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §10.11 and Ch.10

For complex $\nu$ replace $\nu$ by $\overline{\nu }$ on the right-hand sides.

10.10 Continued Fractions10.12 Generating Function and Associated Series

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