About the Project

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

§10.33 Continued Fractions

ⓘ

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		$\frac{I_{\nu}\left(z\right)}{I_{\nu-1}\left(z\right)}=\cfrac{1}{2\nu z^{-1}+}% \cfrac{1}{2(\nu+1)z^{-1}+}\cfrac{1}{2(\nu+2)z^{-1}+}\cdots,$
$z\neq 0$ ,
ⓘ Symbols: $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $z$ : 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		$\frac{I_{\nu}\left(z\right)}{I_{\nu-1}\left(z\right)}=\cfrac{\frac{1}{2}z/\nu}% {1+}\cfrac{\frac{1}{4}z^{2}/(\nu(\nu+1))}{1+}\cfrac{\frac{1}{4}z^{2}/((\nu+1)(% \nu+2))}{1+}\cdots,$
$\nu\neq 0,-1,-2,\dotsc$ .
ⓘ Symbols: $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $z$ : 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

© 2010–2026 NIST / Disclaimer / Feedback; Version 1.2.7; Release date 2026-06-15.

NIST

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov