§13.22 Zeros

Notes:: For (13.22.1) see Olver (1997b, Chapter 12, (7.05)).
Keywords:: Whittaker functions, inequalities
Permalink:: http://dlmf.nist.gov/13.22

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:: $\mathop{O\/}\nolimits\!\left(x\right)$ : 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

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).