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DLMF: §10.11 Analytic Continuation ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions

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10 Bessel Functions Bessel and Hankel Functions 10.10 Continued Fractions 10.12 Generating Function and Associated Series

§10.11 Analytic Continuation

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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		$J_{\nu}\left(ze^{m\pi i}\right)=e^{m\nu\pi i}J_{\nu}\left(z\right),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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		$Y_{\nu}\left(ze^{m\pi i}\right)=e^{-m\nu\pi i}Y_{\nu}\left(z\right)+2i\sin% \left(m\nu\pi\right)\cot\left(\nu\pi\right)J_{\nu}\left(z\right).$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cot\NVar{z}$ : cotangent function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : 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		$\sin\left(\nu\pi\right){H^{(1)}_{\nu}}\left(ze^{m\pi i}\right)=-\sin\left((m-1% )\nu\pi\right){H^{(1)}_{\nu}}\left(z\right)-e^{-\nu\pi i}\sin\left(m\nu\pi% \right){H^{(2)}_{\nu}}\left(z\right),$
ⓘ Symbols: ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : 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, $\sin\NVar{z}$ : 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		$\sin\left(\nu\pi\right){H^{(2)}_{\nu}}\left(ze^{m\pi i}\right)=e^{\nu\pi i}% \sin\left(m\nu\pi\right){H^{(1)}_{\nu}}\left(z\right)+\sin\left((m+1)\nu\pi% \right){H^{(2)}_{\nu}}\left(z\right).$
ⓘ Symbols: ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : 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, $\sin\NVar{z}$ : 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		$\displaystyle{H^{(1)}_{\nu}}\left(ze^{\pi i}\right)$	$\displaystyle=-e^{-\nu\pi i}{H^{(2)}_{\nu}}\left(z\right),$
10.11.5		$\displaystyle{H^{(2)}_{\nu}}\left(ze^{-\pi i}\right)$	$\displaystyle=-e^{\nu\pi i}{H^{(1)}_{\nu}}\left(z\right).$
ⓘ Symbols: ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : 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		$Y_{n}\left(ze^{m\pi i}\right)=(-1)^{mn}(Y_{n}\left(z\right)+2imJ_{n}\left(z% \right)),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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		${H^{(1)}_{n}}\left(ze^{m\pi i}\right)=(-1)^{mn-1}((m-1){H^{(1)}_{n}}\left(z% \right)+m{H^{(2)}_{n}}\left(z\right)),$
ⓘ Symbols: ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : 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		${H^{(2)}_{n}}\left(ze^{m\pi i}\right)=(-1)^{mn}(m{H^{(1)}_{n}}\left(z\right)+(% m+1){H^{(2)}_{n}}\left(z\right)).$
ⓘ Symbols: ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : 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		$\displaystyle J_{\nu}\left(\overline{z}\right)$	$\displaystyle=\overline{J_{\nu}\left(z\right)},$	$\displaystyle Y_{\nu}\left(\overline{z}\right)$	$\displaystyle=\overline{Y_{\nu}\left(z\right)},$
10.11.9		$\displaystyle{H^{(1)}_{\nu}}\left(\overline{z}\right)$	$\displaystyle=\overline{{H^{(2)}_{\nu}}\left(z\right)},$	$\displaystyle{H^{(2)}_{\nu}}\left(\overline{z}\right)$	$\displaystyle=\overline{{H^{(1)}_{\nu}}\left(z\right)}.$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\overline{\NVar{z}}$ : 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 Fractions 10.12 Generating Function and Associated Series

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