The Linear Fractional Transformation Of A Complex Variable: w=(az/b)/(cs/d)

Abstract

A complex number is represented in rectangular coordinates as x/iy, where x and y are real. It is also represented as r(cos e/i sin e) in polar coordinates, where r and e are real. r is called the absolute value, or modulus of the complex number, and e its amplitude. Graphically, a complex number can be represented as a pt. (x,y) or (r,e) in the complex plane. It can also be thought of as a vector from the origin to the pt. (r,e), with length r and angle of inclination e. The complex numbers form a field, and hence possess all the properties of a field. Only two of the properties of the complex numbers will be stated here: (1) IF z=r(cos e/i sin e) , z'=r'(cos e'/i sin e') then, zz'=rr'[cos(e/e')/i sin(e/e')] (2) Let z be the complex conjugate of z. Geometrically, z is obtained by reflecting z upon the x-axis.