10 Bessel Functions Bessel and Hankel Functions 10.9 Integral Representations 10.11 Analytic Continuation

§10.10 Continued Fractions

Notes:: See Watson (1944, §§5.6, 9.65).
Keywords:: Bessel functions, continued fractions
Referenced by:: ¶ ‣ §3.10(ii)
Permalink:: http://dlmf.nist.gov/10.10

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

10.10.1 $\frac{\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)}{\mathop{J_{{\nu-1}}\/}% \nolimits\!\left(z\right)}=\cfrac{1}{2\nu z^{{-1}}-\cfrac{1}{2(\nu+1)z^{{-1}}-% \cfrac{1}{2(\nu+2)z^{{-1}}-\cdots}}},$ $z\neq 0$ ,

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.1.73
Referenced by:: §10.33, §10.74(v)
Permalink:: http://dlmf.nist.gov/10.10.E1
Encodings:: TeX, pMML, png

10.10.2 $\frac{\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)}{\mathop{J_{{\nu-1}}\/}% \nolimits\!\left(z\right)}=\cfrac{\tfrac{1}{2}z/\nu}{1-\cfrac{\tfrac{1}{4}z^{2% }/(\nu(\nu+1))}{1-\cfrac{\tfrac{1}{4}z^{2}/((\nu+1)(\nu+2))}{1-\cdots}}},$ $\nu\neq 0,-1,-2,\ldots$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.1.73
Referenced by:: §10.33, §10.74(v)
Permalink:: http://dlmf.nist.gov/10.10.E2
Encodings:: TeX, pMML, png

See also Cuyt et al. (2008, pp. 349–356).

© 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. 10.9 Integral Representations 10.11 Analytic Continuation