I'm trying to find a weight function w(x) that makes this integral 0
$\int_0^1 w(x) \cosh((\alpha_n+i\omega_n)x) \cosh((\alpha_n-i\omega_n)x)=0$, where where $\omega_n=(2n+1)\frac{\pi}{2}$ and $\tanh(\alpha_n)=1-\frac{J_1(0.002\omega_n)}{0.002\omega_n}$. Anybody have an idea of how to proceed? I've been reading about Fredholm equations. Is that the right track?