# §13.22 Zeros

From (13.14.2) and (13.14.3) $\mathop{M_{\kappa,\mu}\/}\nolimits\!\left(z\right)$ has the same zeros as $\mathop{M\/}\nolimits\!\left(\tfrac{1}{2}+\mu-\kappa,1+2\mu,z\right)$ and $\mathop{W_{\kappa,\mu}\/}\nolimits\!\left(z\right)$ has the same zeros as $\mathop{U\/}\nolimits\!\left(\tfrac{1}{2}+\mu-\kappa,1+2\mu,z\right)$, 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 $\mathop{M_{\kappa,\mu}\/}\nolimits\!\left(z\right)$ is given by

 13.22.1 $\phi_{r}=\frac{j_{2\mu,r}^{2}}{4\kappa}+j_{2\mu,r}\mathop{O\/}\nolimits\!\left% (\kappa^{-\frac{3}{2}}\right),$ Symbols: $\mathop{O\/}\nolimits\!\left(\NVar{x}\right)$: 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

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