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 $I_{\nu}\left(ze^{m\pi i}\right)=e^{m\nu\pi i}I_{\nu}\left(z\right),$

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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, : integer, : 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 $K_{\nu}\left(ze^{m\pi i}\right)=e^{-m\nu\pi i}K_{\nu}\left(z\right)-\pi i\sin% \left(m\nu\pi\right)\csc\left(\nu\pi\right)I_{\nu}\left(z\right).$

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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, : integer, : 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 $I_{\nu}\left(ze^{m\pi i}\right)=(i/\pi)\left(\pm e^{m\nu\pi i}K_{\nu}\left(ze^% {\pm\pi i}\right)\mp e^{(m\mp 1)\nu\pi i}K_{\nu}\left(z\right)\right),$

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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, : integer, : 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 $K_{\nu}\left(ze^{m\pi i}\right)=\csc\left(\nu\pi\right)\left(\pm\sin\left(m\nu% \pi\right)K_{\nu}\left(ze^{\pm\pi i}\right)\mp\sin\left((m\mp 1)\nu\pi\right)K% _{\nu}\left(z\right)\right).$

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