Transformations of Selected Regions of the Z-Plane

The concept of mapping, or transformation of points is very important in the general theory of functions of a complex variable. In this paper transformations will be used to set up correspondences between regions of two complex planes, the z-plane and the w-plane. “Any given correspondence will be unique or one-to-one between points in the two regions if there is just one point in the transformed region corresponding to a given point in the original region and conversely [11, p. 67]” Care must be exercised in the use of the term “mapping” as it is justifiable only when there is a one-to-one correspondence between the two regions. Most books on complex variables include only a short section or a chapter on mapping. The purpose of this paper is to present to the reader a more complete picture of various transformations through illustrated mappings. This will be accomplished through a study of the behavior of certain regions under a few simple transformations. Only specific regions will be considered as the general cases become complicated and are beyond the scope of this paper. The regions will be limited to rectangles, triangles, semi-infinite and infinite strips, and the unit circle. The transformations w = ez, w = sin z, w = cos z, w = sinh z 9 and w = cosh z will be considered. The last three of these transformations will be studied in Chapter V by using the idea of successive transformations. Such graphic terms as translation, rotation, and expansion will be used