Singular integral equations with a Cauchy kernel (Q1074813)

The author proposes a method to obtain closed form expressions for continuously differentiable even solution function \(\phi\) (\(\omega)\) for the linear singular integral equation \[ \phi (\omega)-\frac{\lambda (\omega)}{\pi}\int^{\infty}_{-\infty}\frac{\phi (y)}{\omega - y}dy=f(o) \] where \(\lambda\) (\(\omega)\), f(\(\omega)\) are given continuously differentiable functions such that \(\lambda (-\omega)=- \lambda (\omega)\) and \(f(-\omega)=f(\omega)\). He then generalizes the techniques to obtain closed form expressions for continuously differentiable even solution functions for the nonlinear singular integral equation \(-\frac{1}{\pi^ 2}(\int^{\infty}_{- \infty}\frac{\phi (y)}{\omega -y}dy)^ 2+\frac{2}{p}\cdot \lambda (\omega)\cdot \phi (\omega)\cdot \int^{\infty}_{-\infty}\frac{\phi (y)}{\omega -y}dy+\frac{1}{\quad \pi}k(\omega)\cdot \lambda (\omega)\cdot \int^{\infty}_{-\infty}\frac{\phi (y)}{\omega -y}dy+\phi^ 2(\omega)\cdot k\) \((\omega)\cdot \phi (\omega)=f(\omega)\), given continuously differentiable functions: odd \(\lambda\) (\(\omega)\), even f(\(\omega)\), k(\(\omega)\). Some examples of linear and nonlinear singular equations are also provided.