On the solvability in a closed form of a class of singular integral equations (Q1409799)

The authors investigate methods of solution for singular integral equations as \[ \sum^n_{k=1}a_k(t)\varphi(\varepsilon_k t)+ \sum^n_{k=1}\frac{1}{\pi i}\int_\Gamma \frac{\tau^{n-1-k}t^k}{\tau^n-t^n}m_k(\tau,t) \varphi(\tau) d\tau=f(t), \tag{1} \] where \(\Gamma\) is the unit circle on the complex plane, \(\varepsilon_k=exp(2\pi i k/k),\) \(k=0,1,\ldots,n.\) Equation (1) is reduced to a Riemann value boundary problem. They prove that equation (1) has a solution if the corresponding Riemann boundary value problem has a solution.