Singular integral equation

Summary

A singular integral equation is an integral equation whose kernel has a strong singularity. The integral does not converge absolutely, and it is taken as a Cauchy principal value. The main case is the one-dimensional equation on a curve with the Cauchy kernel $1/(\tau - t)$. Its integral operator is not compact, so the Fredholm theory does not apply directly. Instead, the theory of Noether, Muskhelishvili and others gives an index that measures the failure of the Fredholm alternative. Such equations arise in elasticity, aerodynamics, fracture mechanics and boundary value problems for holomorphic functions.