§13.22 Zeros

From (13.14.2) and (13.14.3) $M_{\kappa ,\mu }\left(z\right)$ has the same zeros as $M(\frac{1}{2}+\mu -\kappa ,1+2\mu ,z)$ and $W_{\kappa ,\mu }\left(z\right)$ has the same zeros as $U(\frac{1}{2}+\mu -\kappa ,1+2\mu ,z)$, 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 $\mu (\ge 0)$ is fixed and $\kappa (>0)$ is large, then the $r$th positive zero ${\varphi }_r$ of $M_{\kappa ,\mu }\left(z\right)$ is given by

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).