The computation of all the zeros of a holomorphic function \(f\) that lie inside the unit circle is performed via numerical evaluation by trapezoidal rule of certain Cauchy integral. The above integral contains, in principle, the logarithmic derivative \(f'/f\) which has the poles at each zero of \(f\). The substantial simplification of proposed method consists in replacing of the above integral by an integral containing only \(1/f\), the dervative \(f'\) being no longer needed. The location of the zeros is achieved via the computation of the zeros of formal orthogonal polynomials starting from a generalized eigenvalue problem. An error analysis is presented.