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6000 DLMF: §10.10 Continued Fractions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions

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

§10.10 Continued Fractions

ⓘ

Keywords:: Bessel functions, continued fractions
Notes:: See Watson (1944, §§5.6, 9.65).
Referenced by:: §3.10(ii)
Permalink:: http://dlmf.nist.gov/10.10
See also:: Annotations for Ch.10

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

10.10.1		$\frac{J_{\nu}\left(z\right)}{J_{\nu-1}\left(z\right)}=\cfrac{1}{2\nu z^{-1}-% \cfrac{1}{2(\nu+1)z^{-1}-\cfrac{1}{2(\nu+2)z^{-1}-\cdots}}},$
$z\neq 0$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $z$ : 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 See also: Annotations for §10.10 and Ch.10

10.10.2		$\frac{J_{\nu}\left(z\right)}{J_{\nu-1}\left(z\right)}=\cfrac{\tfrac{1}{2}z/\nu% }{1-\cfrac{\tfrac{1}{4}z^{2}/(\nu(\nu+1))}{1-\cfrac{\tfrac{1}{4}z^{2}/((\nu+1)% (\nu+2))}{1-\cdots}}},$
$\nu\neq 0,-1,-2,\dotsc$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $z$ : 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: Annotations for §10.10 and Ch.10

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

10.9 Integral Representations 10.11 Analytic Continuation

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