Note on the complex zeros of \(H^{\prime}_{\nu}+\mathrm i\zeta H_{\nu}(x)=0\) (Q869518)

This letter deals with a method for determining the complex \(\nu\)-zeros of the expression \[ H_\nu '(x)+i\zeta H_\nu(x)=0, \] where \(H_\nu(x)\) denotes the Hankel function of the first kind, arising in scattering problems from cylinders. The variable \(x\) (corresponding to the frequency) is supposed to be a large positive quantity while the parameter \(\zeta\) is complex and results from the treatment when an impedance boundary condition is imposed. An approach expressing the zeros of the above Hankel functions in terms of the zeros of the Airy function and its derivative leads to approximations for the zeros of (1) that are satisfactory outside the zone given by \((x/2)^{1/3}|\zeta|\simeq 1\). An alternative method that avoids this difficulty uses asymptotic expansions for the Hankel function in terms of only one type of Airy function Ai in the approximate form \[ \text{Ai}'(-z)-q\text{Ai} (z)=0, \] where \(z\) and \(q\) are appropriate quantities that depend on \(x\), \(\nu\) and \(\zeta\). A numerical illustration of the accuracy obtainable for the zeros of (1) is given.