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

§10.33 Continued Fractions

Notes:: Combine (10.10.1), (10.10.2) with (10.27.6).
Keywords:: continued fractions, modified Bessel functions
Referenced by:: ¶ ‣ §3.10(ii)
Permalink:: http://dlmf.nist.gov/10.33

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

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

Symbols:: $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, : complex variable and $\nu$ : complex parameter
Referenced by:: §10.74(v)
Permalink:: http://dlmf.nist.gov/10.33.E1
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, : 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 Cuyt et al. (2008, pp. 361–367).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 10.32 Integral Representations 10.34 Analytic Continuation