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

If $m$ is a large positive integer, then

 10.70.2 zeros of $\mathop{\mathrm{ber}_{\nu}\/}\nolimits x$ $\displaystyle\sim\sqrt{2}(t-f(t)),$ $t=(m-\tfrac{1}{2}\nu-\tfrac{3}{8})\pi$, zeros of $\mathop{\mathrm{bei}_{\nu}\/}\nolimits x$ $\displaystyle\sim\sqrt{2}(t-f(t)),$ $t=(m-\tfrac{1}{2}\nu+\tfrac{1}{8})\pi$, zeros of $\mathop{\mathrm{ker}_{\nu}\/}\nolimits x$ $\displaystyle\sim\sqrt{2}(t+f(-t)),$ $t=(m-\tfrac{1}{2}\nu-\tfrac{5}{8})\pi$, zeros of $\mathop{\mathrm{kei}_{\nu}\/}\nolimits x$ $\displaystyle\sim\sqrt{2}(t+f(-t)),$ $t=(m-\tfrac{1}{2}\nu-\tfrac{1}{8})\pi$.

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).