We propose a method to search all the zeros of a complex function in a given compact region D Cn. The function f: Cn Cn to be considered is assumed to consist of polynomial and transcendental terms and to satisfy f (x) Rn for any x Rn. Using the properties of such a complex function, we can compute the number of zeros and determine the starting points of paths on the boundary of D, which attain all the zeros of f in D without encountering a singular point. A piecewiselinear approximation of the function on a triangulation is used for both computing the number of zeros and following the paths.