On the Solvability in a Closed Form of a Class of Singular Integral Equations

Nguyen Van  Mau and Nguyen Minh Tuan


    This paper deals with the solvability of singular integral equations of the form\vs\vs  $$\sum_{k=1}^na_k(t) \varphi(\epsilon_kt) + \sum_{k=1}^n \frac{1}{\pii} \int\limits_{\Gamma}\frac{\tau^{n-1-k}t^k}{ \tau^n-t^n} m_k(\tau,t) \varphi(\tau)d\tau  =  f(t),\eqno(1)$$   where $\Gamma$ is the unit circle on the complex   plane, $\epsilon_1=   \exp(2\pi i/n), \epsilon_k=\epsilon_1^k.$ We indicate   that the   equation (1) is exactly a singular integral equation   of Cauchy type with a   certain shift. The method in this paper is to   reduce the equation (1) to the  well-known Riemann   boundary value  problems and describe  solutions  in a closed form.