Singular Integral Equations Boundary Problems Of Function Theory And Their Application To Mathematical Physics N I Muskhelishvili __exclusive__ Official

Muskhelishvili transformed this into a singular integral equation of the second kind:

One of the central pillars of the text is the rigorous treatment of the Riemann-Hilbert problem. In simple terms, this is the problem of finding an analytic function within a domain given a linear relationship between its real and imaginary parts on the boundary.

The kernel (1/(\tau - t)) becomes infinite at (\tau = t) at such a rate that the ordinary Lebesgue integral diverges. However, the principal value exists for Hölder-continuous functions (\phi(t)). This singular nature is not a nuisance—it encodes the physical behavior of field variables near edges, cracks, or boundaries.

This article explores the mathematical structure of singular integral equations, their connection to boundary value problems of analytic function theory, and their profound applications to mathematical physics.