Inversion of a logarithmic operator defined on a regular set of arcs lying on a circle

The theory of singular integral equations is used to derive simple inversion formulas for a logarithmic operator defined on a contour consisting of an arbitrary number of identical arcs lying on a circle at an equal angular spacing. The action of the inverse operator on trigonometric functions is calculated, and the moments of the inverse operator with trigonometric functions are found. Even simpler formulas are derived in the approximation of small arcs.