10 Bessel FunctionsModified Bessel Functions10.33 Continued Fractions10.35 Generating Function and Associated Series

§10.34 Analytic Continuation

ⓘ

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		$$I_{\nu }\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={\mathrm{e}}^{m\nu \pi \mathrm{i}}{I}_{\nu }\left(z\right),$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: 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		$$K_{\nu }\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={\mathrm{e}}^{-m\nu \pi \mathrm{i}}{K}_{\nu }\left(z\right)-\pi \mathrm{i}\sin \left(m\nu \pi \right)\csc \left(\nu \pi \right)I_{\nu }\left(z\right).$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\sin z$: 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		$I_{\nu }\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)$	$=(\mathrm{i}/\pi )\left(\pm {\mathrm{e}}^{m\nu \pi \mathrm{i}}{K}_{\nu }\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)\mp {\mathrm{e}}^{(m\mp 1)\nu \pi \mathrm{i}}{K}_{\nu }\left(z\right)\right),$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: 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		$K_{\nu }\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)$	$=\csc \left(\nu \pi \right)\left(\pm \sin \left(m\nu \pi \right)K_{\nu }\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)\mp \sin \left((m\mp 1)\nu \pi \right)K_{\nu }\left(z\right)\right).$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\sin z$: 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		$$K_n\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={(-1)}^{mn}{K}_n\left(z\right)+{(-1)}^{n(m-1)-1}m\pi \mathrm{i}{I}_n\left(z\right),$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: 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		$$K_n\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)=\pm {(-1)}^{n(m-1)}m{K}_n\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)\mp {(-1)}^{nm}(m\mp 1)K_n\left(z\right).$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $K_{\nu }\left(z\right)$: 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		$I_{\nu }\left(\overline{z}\right)$	$=\overline{I_{\nu }\left(z\right)},$
		$K_{\nu }\left(\overline{z}\right)$	$=\overline{K_{\nu }\left(z\right)}.$
ⓘ Symbols: $\overline{z}$: complex conjugate, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: 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 Fractions10.35 Generating Function and Associated Series

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