On the solution of a special kind of singular integral equation
While using the inversion formula for a Cauchy type singular integral equation to approximate numerically the solution, the author deals with the case where the solution is not continuously differentiable, but is in a special class of Hölder continuous functions, \(C^\alpha(L,\mathbb{C})\) on a simple, closed smooth curve in \(\mathbb{C}\) with the usual norm \(|\cdot|_{C^\alpha(L,\mathbb{C})}\). The integral equation is \[ {1\over\pi i} \int_L {\phi(t)\over t-t_0} dt= \psi(t_0),\quad t_0\in L \] and its solution is \[ \phi(t_0)= {1\over\pi i} \int_L {\psi(t)\over t-t_0} dt,\quad t_0\in L, \] where \(\phi\in C^\alpha([0,2\pi],\mathbb{C})\). Various sections of the paper give appropriate concepts, definitions and formulae, so employed.
scientific article; zbMATH DE number 1029188