 Solution

For fixed $k$ and $\ell$, the line $-z/\kappa$ becomes increasingly negative, so each intersection approaches its branch’s left-hand vertical asymptote. Write $z_n=a_n+{\epsilon }_n$, where $a_n=(n-1/2)\pi$ and ${\epsilon }_n>0$. The eigenvalue equation givesSince $\cot \epsilon ={\epsilon }^{-1}-\epsilon /3+O({\epsilon }^3)$, it follows that ${\epsilon }_n=\kappa /a_n+O(a_n^{-3})$. ThusThe half-integer phase identifies the sharper high-frequency behaviour of these Dirichlet-Robin eigenvalues on a finite interval. The expansion holds with the Robin coefficient fixed; it is not a uniform expansion in $k\to \infty$.