Finite Hilbert Transforms Logarithmic Potentials and Singular Integral Equations

TL;DR

This paper explores finite Hilbert transforms and logarithmic integrals, providing formulas that aid in solving singular integral equations and analyzing equilibrium measures in random matrix models.

Contribution

It introduces new formulas for finite Hilbert transforms and logarithmic integrals with applications in equilibrium measures and random matrix theory.

Findings

Derived formulas for finite Hilbert transforms and logarithmic integrals.

Applied these formulas to determine equilibrium measures.

Solved singular integral equations using the developed methods.

Abstract

Several interesting formulas concerning finite Hilbert transform and logarithmic integrals are proved with application in determining equilibrium measures, planar limits of analytic random matrix models with $1-$cut potential and solving singular integral equations.