§10.70 Zeros

Notes:: Revert (10.68.18) and (10.68.21) (§2.2).
Keywords:: Kelvin functions
Permalink:: http://dlmf.nist.gov/10.70

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

10.70.1 $\frac{\mu-1}{16t}+\frac{\mu-1}{32t^{2}}+\frac{(\mu-1)(5\mu+19)}{1536t^{3}}+\frac{3(\mu-1)^{2}}{512t^{4}}+\cdots.$

Symbols:: $f(t)$ : function
A&S Ref:: 9.10.35
Referenced by:: §10.70
Permalink:: http://dlmf.nist.gov/10.70.E1
Encodings:: TeX, pMML, png

If $m$ is a large positive integer, then

10.70.2

$\mbox{zeros of $\mathop{\mathrm{ber}_{{\nu}}\/}\nolimits x$}\sim\sqrt{2}(t-f(t)),$ $t=(m-\tfrac{1}{2}\nu-\tfrac{3}{8})\pi$ ,

$\mbox{zeros of $\mathop{\mathrm{bei}_{{\nu}}\/}\nolimits x$}\sim\sqrt{2}(t-f(t)),$ $t=(m-\tfrac{1}{2}\nu+\tfrac{1}{8})\pi$ ,

$\mbox{zeros of $\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits x$}\sim\sqrt{2}(t+f(-t)),$ $t=(m-\tfrac{1}{2}\nu-\tfrac{5}{8})\pi$ ,

$\mbox{zeros of $\mathop{\mathrm{kei}_{{\nu}}\/}\nolimits x$}\sim\sqrt{2}(t+f(-t)),$ $t=(m-\tfrac{1}{2}\nu-\tfrac{1}{8})\pi$ .

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).