Eigenfunction and their orthogonality with respect to the weight function The Eigenfunction and their orthogonality with respect to the weight function $\sigma$ is defined as $$\int_{a}^{b}\phi _n\text{(x)}\phi _m\text{(x)$\sigma $(x)dx=0}$$ Given that I have some function $$ g(x)=-\sum_{n=1}^{\infty}{\beta_n}\tanh\left( \frac{n\pi L}{H} \right)\sin\left( \frac{n\pi y}{H} \right) $$ How should I go about finding what $$\beta_n$$ is using the properties of the eigenfunction and their orthogonality? This is in the context of PDEs and the Sturm-Lioville problem. Any help is appreciated!

My assumption is that you have found this expansion by solving the sturm louville equation with weight function $\sigma =1$. In which case, the $\tanh(\ldots)\sinh(\ldots)$ are the riven functions, since they are orthonormal, taking an inner product with the $m$-th eigenfunction would yield $$\beta_m=-\int_0^Lg(x)\tanh(m\pi L/H)\sin(m\pi L/H)\,dx.$$ If the weight function is not $1$ it would also sit in the integral.