Application of the Gakhov–Muskhelishvili Formula for Finding the Zeros of Piecewise Analytic Functions

We consider piecewise analytic functions $$\mathscr{F}(z)$$ with finitely many complex zeros and with line of discontinuinity coinciding with the real axis. For this type of functions, the paper presents an explicit representation of Gakhov–Muskhelishvili type known in the theory of the Riemann linear conjugation problem for analytic functions. The obtained representation reduces the problem of calculation of zeros of the function $$\mathscr{F}(z)$$ to finding the zeros of an explicitly written polynomial. We apply the result to a function which arises in study of effect of an electric field on a plasma layer.