Abstract: In this paper, a new simple method is presented. Owing to this method, the fractal images of the upper half plane, the square limit and the circle limit in the dynamic plane and the parameter plane are constructed from the complex mapping z ←e iπ2 z\+w +c（w=α+ i β,α,β∈R\+1） . The dynamic plane and the parameter plane are treated by means of periodization. The fundamental compute region B is determined and self\|similarly mapped into the upper half plane by using the homeomorphic affine transformation θ , so the fractal images of the upper half plane, square limit, and circle limit can be constructed. A great variety of these kinds of the J and M fractal artistic patterns from the complex mapping z ←e iπ2 z\+w +c can be created with the method presented.