[ X(z) = \exp\left \frac12\pi i \int_L \frac\ln G(t)t - z dt \right ]
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. [ X(z) = \exp\left \frac12\pi i \int_L \frac\ln
The solution (\sigma(x) = C/\sqrta^2 - x^2) (the edge singularity) is immediate from Muskhelishvili’s theory—a result that predates, but is rigorously justified by, his methods. [ \Phi(z) = \frac12\pi i \int_L \frac\phi(\tau)\tau -
[ \Phi(z) = \frac12\pi i \int_L \frac\phi(\tau)\tau - z d\tau, \quad z \notin L ] $\phi(z)$ and $\psi(z)$
[ \Phi^+(t) = G(t) \Phi^-(t) + g(t) ]
Perhaps the most famous application discussed in the book is the solution of the biharmonic equation for plane elasticity. The stress and displacement in a two-dimensional elastic body can be expressed in terms of two complex potentials, $\phi(z)$ and $\psi(z)$, known as the .