The authors present a combination of global iterative methods, based on the Fatou-Julia theory, and local methods to find selected roots of elementary transcendental equations, \(z=F(z,c)\), \(z\) and \(c\) complex, that occur in different applications. Suitable starting values for the iteration function, \(z_{n+1}=F(z_ n,c)\), and appropriate regions for the determination of the inverse iteration function, \(z_{n+1}=F^{- 1}(z_ n,c)\), are given. Convergence criteria are derived. Certain fixed points of \(F\) can be reached quickly by means of a local method, like Newton's method. It is shown that Newton's method leads to attractive cycles or to unpredictable roots when the starting values are near the Fatou-Julia set.