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DLMF: §10.34 Analytic Continuation ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions

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10 Bessel Functions Modified Bessel Functions 10.33 Continued Fractions 10.35 Generating Function and Associated Series

§10.34 Analytic Continuation

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Keywords:: analytic continuation, modified Bessel functions
Notes:: See Watson (1944, p. 80) and Olver (1997b, pp. 253, 381). For (10.34.3) take $m=\pm 1$ in (10.34.2), and combine with (10.34.1).
Referenced by:: §10.27, Erratum (V1.0.19) for Notation
Permalink:: http://dlmf.nist.gov/10.34
See also:: Annotations for Ch.10

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

10.34.1		$I_{\nu}\left(ze^{m\pi i}\right)=e^{m\nu\pi i}I_{\nu}\left(z\right),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $m$ : integer, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.6.30 Referenced by: §10.30(ii), §10.34, §10.47(v) Permalink: http://dlmf.nist.gov/10.34.E1 Encodings: TeX, pMML, png See also: Annotations for §10.34 and Ch.10

10.34.2		$K_{\nu}\left(ze^{m\pi i}\right)=e^{-m\nu\pi i}K_{\nu}\left(z\right)-\pi i\sin% \left(m\nu\pi\right)\csc\left(\nu\pi\right)I_{\nu}\left(z\right).$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$ : cosecant function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\sin\NVar{z}$ : sine function, $m$ : integer, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.6.31 Referenced by: §10.27, §10.34, §10.34 Permalink: http://dlmf.nist.gov/10.34.E2 Encodings: TeX, pMML, png See also: Annotations for §10.34 and Ch.10

10.34.3		$\displaystyle I_{\nu}\left(ze^{m\pi i}\right)$	$\displaystyle=(i/\pi)\left(\pm e^{m\nu\pi i}K_{\nu}\left(ze^{\pm\pi i}\right)% \mp e^{(m\mp 1)\nu\pi i}K_{\nu}\left(z\right)\right),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $m$ : integer, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.34, §10.40(i), §10.40(iii), §10.41(iv) Permalink: http://dlmf.nist.gov/10.34.E3 Encodings: TeX, pMML, png See also: Annotations for §10.34 and Ch.10
10.34.4		$\displaystyle K_{\nu}\left(ze^{m\pi i}\right)$	$\displaystyle=\csc\left(\nu\pi\right)\left(\pm\sin\left(m\nu\pi\right)K_{\nu}% \left(ze^{\pm\pi i}\right)\mp\sin\left((m\mp 1)\nu\pi\right)K_{\nu}\left(z% \right)\right).$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$ : cosecant function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\sin\NVar{z}$ : sine function, $m$ : integer, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.34, §10.40(i) Permalink: http://dlmf.nist.gov/10.34.E4 Encodings: TeX, pMML, png See also: Annotations for §10.34 and Ch.10

If $\nu=n(\in\mathbb{Z})$ , then limiting values are taken in (10.34.2) and (10.34.4):

10.34.5		$K_{n}\left(ze^{m\pi i}\right)=(-1)^{mn}K_{n}\left(z\right)+(-1)^{n(m-1)-1}m\pi iI% _{n}\left(z\right),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $m$ : integer, $n$ : integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/10.34.E5 Encodings: TeX, pMML, png See also: Annotations for §10.34 and Ch.10

10.34.6		$K_{n}\left(ze^{m\pi i}\right)=\pm(-1)^{n(m-1)}mK_{n}\left(ze^{\pm\pi i}\right)% \mp(-1)^{nm}(m\mp 1)K_{n}\left(z\right).$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $m$ : integer, $n$ : integer and $z$ : complex variable Referenced by: §10.40(i) Permalink: http://dlmf.nist.gov/10.34.E6 Encodings: TeX, pMML, png See also: Annotations for §10.34 and Ch.10

For real $\nu$ ,

10.34.7		$\displaystyle I_{\nu}\left(\overline{z}\right)$	$\displaystyle=\overline{I_{\nu}\left(z\right)},$
10.34.7		$\displaystyle K_{\nu}\left(\overline{z}\right)$	$\displaystyle=\overline{K_{\nu}\left(z\right)}.$
ⓘ Symbols: $\overline{\NVar{z}}$ : complex conjugate, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.6.32 Permalink: http://dlmf.nist.gov/10.34.E7 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.34 and Ch.10

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

10.33 Continued Fractions 10.35 Generating Function and Associated Series

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