ABSTRACT
Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 4.5 Necessary Conditions for w = f(z) to Represent Conformal
Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 4.6 Superficial Magnification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 4.7 Some Elementary Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 4.8 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 4.9 Bilinear or Mo¨bius Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 4.10 Product or Resultant of Two Bilinear Transformations . . . . . . . . . 194 4.11 Every Bilinear Transformation is the Resultant of Elementary
Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 4.12 Bilinear Transformation as the Resultant of an Even Number of
Inversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 4.13 The Linear Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 4.14 Cross-Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 4.15 Preservation of Cross-Ratio under Bilinear Transformations . . . . 201 4.16 Preservation of the Family of Circles and Straight Lines under
Bilinear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 4.17 Two Important Families of Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 4.18 Fixed Point of Bilinear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 214 4.19 Normal Form of a Bilinear Transformation . . . . . . . . . . . . . . . . . . . . . . 215 4.20 Elliptic, Hyperbolic, and Parabolic Transformations . . . . . . . . . . . . 218 4.21 Special Bilinear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 4.22 Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
If w = f(z) is an analytic function of z defined in a domain D of the z-plane, then, to every value of z in D, there corresponds a unique value of w, called the image of the said value of z, which we may represent in another complex plane called the w-plane in contrast to the representation of the real-valued
function where the images are represented in the same plane. We then say that the points of the domain D in the z-plane are mapped into corresponding points of the w-plane and the set of points of the w-plane which are images of the points of D forms the map of D under the transformation w = f(z). Some information about the function can, however, be displayed by representing sets of corresponding points z = x + iy and w = u + iv on their respective planes. The defining equations are
u = u(x, y), v = v(x, y). (4.1)
The correspondence defined by equations (4.1) between the points in the z-plane and w-plane is called a mapping or transformation of points in the z-plane into points of the w-plane by the function f . The corresponding sets of points in the two planes are called images of each other. The equations (4.1) are called transformations. If, to each point of the z-plane, there corresponds one and only one point of the w-plane and, conversely, we say that the correspondence is one-to-one.
