FiniteHilbertTransform.jl
Computation of Hilbert Transform with finite boundaries and Landau's prescription.
While both Legendre and Chebyshev methods are implemented, we recommend using Legendre techniques for integration.
FiniteHilbertTransform.jl's core functionality precomputes the Hilbert-transformed Legendre functions $Q_k(w)$ for a given complex frequency $\omega$. The Hilbert transform is defined as $Q_k(w) = \int_{-1}^{1} \frac{P_k(u)}{u - w} du$, where $P_k(u)$ is the Legendre function of the first kind. It is important to note that $Q_k(w) = -2 q_k(w)$ for real values of $w$, where $q_k(w)$ represents the Legendre functions of the second kind.