# §13.22 Zeros

From (13.14.2) and (13.14.3) $M_{\kappa,\mu}\left(z\right)$ has the same zeros as $M\left(\tfrac{1}{2}+\mu-\kappa,1+2\mu,z\right)$ and $W_{\kappa,\mu}\left(z\right)$ has the same zeros as $U\left(\tfrac{1}{2}+\mu-\kappa,1+2\mu,z\right)$, hence the results given in §13.9 can be adopted.

Asymptotic approximations to the zeros when the parameters $\kappa$ and/or $\mu$ are large can be found by reversion of the uniform approximations provided in §§13.20 and 13.21. For example, if $\mu(\geq 0)$ is fixed and $\kappa(>0)$ is large, then the $r$th positive zero $\phi_{r}$ of $M_{\kappa,\mu}\left(z\right)$ is given by

 13.22.1 $\phi_{r}=\frac{j_{2\mu,r}^{2}}{4\kappa}+j_{2\mu,r}O\left(\kappa^{-\frac{3}{2}}% \right),$ ⓘ Symbols: $O\left(\NVar{x}\right)$: order not exceeding, $\phi_{r}$: positive zeros and $j_{b,r}$: positive zero of Bessel Referenced by: §13.22, §13.22 Permalink: http://dlmf.nist.gov/13.22.E1 Encodings: TeX, pMML, png See also: Annotations for §13.22 and Ch.13

where $j_{2\mu,r}$ is the $r$th positive zero of the Bessel function $J_{2\mu}\left(x\right)$10.21(i)). (13.22.1) is a weaker version of (13.9.8).