Implement derivative wrt k0
We would like to have n'_\lambda(k_0)
Let F(n, k_0)
be the wronskian so that F(n_\lambda, k_0) = 0
. Then
n'_\lambda = - \frac{\partial F}{\partial k_0} \cdot \left[\frac{\partial F}{\partial n}\right]^{-1}
The missing piece is
\frac{\partial F}{\partial k_0}
Alternatively, we could just compute n'_\lambda(k_0)
via finite difference approximation, i.e.
n'_\lambda = [n_\lambda(k_0 + \Delta) - n_\lambda(k_0)] / \Delta