Logarithmic potentials with external fields

In recent years, approximation theory and orthogonal polynomials have seen a surge in solutions to previously challenging problems, thanks to the integration of approximation techniques with classical potential theory, particularly logarithmic potentials related to polynomials and real line issues. Many applications stem from extending classical logarithmic potential theory to scenarios involving weights or external fields. Recent advancements include the development of non-classical orthogonal polynomials concerning exponential weights, orthogonal polynomials for general measures with compact support, incomplete polynomials and their generalizations, and multipoint Pade approximation. This new approach has led to significant breakthroughs, such as resolving Freud's problems on the asymptotics of orthogonal polynomials with weights like exp(-|x|), the “1/9-th” conjecture on rational approximation of exp(x), and determining the exact asymptotic constant in rational approximation of |x|. This work aims to offer a self-contained introduction to the "weighted" potential theory and its diverse applications while also thoroughly developing the classical theory of logarithmic potentials.