Analytic continuation with respect to a parameter of the Green's functions of exterior boundary value problems for the two-dimensional Helmholtz equation. I

In the first part of the paper one studies the distribution in the half-plane $\{\nu:|{\arg\nu}|<\pi/2\}$ of the roots of the functions $H_\nu'(k)$ and $H_\nu'(k)+igH_\nu(k)$ and of the variable $\nu$ for arbitrary fixed complex $k$ from the region $K(\delta,\varkappa)=\{k:-\delta<\arg k<\pi/2-\delta,\ \varkappa<|k|\}$ for some $\delta\in(0,\pi/2)$ and $\varkappa>0$, where $H_\nu(k)$ is the first Hankel function, $H_\nu'(k)$ is its derivative with respect to $k$, and $g$ is an arbitrary nonnegative number.
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