10 Bessel Functions Kelvin Functions 10.69 Uniform Asymptotic Expansions for Large Order 10.71 Integrals

§10.70 Zeros

ⓘ

Keywords:: Kelvin functions, asymptotic approximations for large zeros, zeros
Notes:: Revert (10.68.18) and (10.68.21) (§2.2).
Permalink:: http://dlmf.nist.gov/10.70
See also:: Annotations for Ch.10

Asymptotic approximations for large zeros are as follows. Let $\mu=4\nu^{2}$ and $f(t)$ denote the formal series

10.70.1		$\frac{\mu-1}{16t}+\frac{\mu-1}{32t^{2}}+\frac{(\mu-1)(5\mu+19)}{1536t^{3}}+% \frac{3(\mu-1)^{2}}{512t^{4}}+\dotsi.$
ⓘ Defines: $f(t)$ : function (locally) A&S Ref: 9.10.35 Referenced by: §10.70 Permalink: http://dlmf.nist.gov/10.70.E1 Encodings: TeX, pMML, png See also: Annotations for §10.70 and Ch.10

If $m$ is a large positive integer, then

10.70.2		zeros of $\operatorname{ber}_{\nu}x$	$\displaystyle\sim\sqrt{2}(t-f(t)),$
	$t=(m-\tfrac{1}{2}\nu-\tfrac{3}{8})\pi$ ,
		zeros of $\operatorname{bei}_{\nu}x$	$\displaystyle\sim\sqrt{2}(t-f(t)),$
	$t=(m-\tfrac{1}{2}\nu+\tfrac{1}{8})\pi$ ,
		zeros of $\operatorname{ker}_{\nu}x$	$\displaystyle\sim\sqrt{2}(t+f(-t)),$
	$t=(m-\tfrac{1}{2}\nu-\tfrac{5}{8})\pi$ ,
		zeros of $\operatorname{kei}_{\nu}x$	$\displaystyle\sim\sqrt{2}(t+f(-t)),$
	$t=(m-\tfrac{1}{2}\nu-\tfrac{1}{8})\pi$ .
ⓘ Symbols: $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $m$ : integer, $x$ : real variable, $\nu$ : complex parameter and $f(t)$ : function A&S Ref: 9.10.36 Referenced by: §10.70 Permalink: http://dlmf.nist.gov/10.70.E2 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §10.70 and Ch.10

In the case $\nu=0$ , numerical tabulations (Abramowitz and Stegun (1964, Table 9.12)) indicate that each of (10.70.2) corresponds to the $m$ th zero of the function on the left-hand side. For the next six terms in the series (10.70.1) see MacLeod (2002a).

10.69 Uniform Asymptotic Expansions for Large Order 10.71 Integrals

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