§10.70 Zeros

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

10.70.1		$$\frac{\mu -1}{16t}+\frac{\mu -1}{32t^2}+\frac{(\mu -1)(5\mu +19)}{1536t^3}+\frac{3{(\mu -1)}^2}{512t^4}+\mathrm{\cdots }.$$
ⓘ 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 ${\mathrm{ber}}_{\nu }x$	$\sim \sqrt{2}(t-f(t)),$
	$t=(m-\frac{1}{2}\nu -\frac{3}{8})\pi$,
		zeros of ${\mathrm{bei}}_{\nu }x$	$\sim \sqrt{2}(t-f(t)),$
	$t=(m-\frac{1}{2}\nu +\frac{1}{8})\pi$,
		zeros of ${\ker }_{\nu }x$	$\sim \sqrt{2}(t+f(-t)),$
	$t=(m-\frac{1}{2}\nu -\frac{5}{8})\pi$,
		zeros of ${\mathrm{kei}}_{\nu }x$	$\sim \sqrt{2}(t+f(-t)),$
	$t=(m-\frac{1}{2}\nu -\frac{1}{8})\pi$.
ⓘ Symbols: ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: 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).