10 Bessel Functions Modified Bessel Functions 10.32 Integral Representations 10.34 Analytic Continuation

§10.33 Continued Fractions

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Keywords:: continued fractions, modified Bessel functions
Notes:: Combine (10.10.1), (10.10.2) with (10.27.6).
Referenced by:: §3.10(ii)
Permalink:: http://dlmf.nist.gov/10.33
See also:: Annotations for Ch.10

Assume $I_{\nu-1}\left(z\right)\neq 0$ . Then

10.33.1 $\frac{I_{\nu}\left(z\right)}{I_{\nu-1}\left(z\right)}=\cfrac{1}{2\nu z^{-1}+}% \cfrac{1}{2(\nu+1)z^{-1}+}\cfrac{1}{2(\nu+2)z^{-1}+}\cdots,$ $z\neq 0$ ,

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Symbols:: $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, : complex variable and $\nu$ : complex parameter
Referenced by:: §10.74(v)
Permalink:: http://dlmf.nist.gov/10.33.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §10.33 and Ch.10

10.33.2 $\frac{I_{\nu}\left(z\right)}{I_{\nu-1}\left(z\right)}=\cfrac{\frac{1}{2}z/\nu}% {1+}\cfrac{\frac{1}{4}z^{2}/(\nu(\nu+1))}{1+}\cfrac{\frac{1}{4}z^{2}/((\nu+1)(% \nu+2))}{1+}\cdots,$ $\nu\neq 0,-1,-2,\dotsc$ .

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Symbols:: $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, : complex variable and $\nu$ : complex parameter
Referenced by:: §10.74(v)
Permalink:: http://dlmf.nist.gov/10.33.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §10.33 and Ch.10

See also Cuyt et al. (2008, pp. 361–367).

10.32 Integral Representations 10.34 Analytic Continuation

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