A modification of the classical quadrature method for locating zeros of analytic functions

The classical method for determination of a simple zero a of an analytic function f ( z ) inside a closed contour C by using the formula a =(2 πi ) −1 ε C [ zf' ( z )/ f ( z )] dz is reconsidered and modified. The modification consists in using the Cauchy theorem instead of the Cauchy formula and, further, evaluating numerically a as one of the zeros of an appropriate polynomial depending on the quadrature rule used. The case of circular contours with application of the trapezoidal rule is considered in detail and numerical results are presented.