The exterior of the unit circle on the s-plane is thus mapped onto the interior of this 

 circle on the ty-plane, and vice versa. Radial lines transform into radial Lines with reflection 

 in the real axis. 



It is easily shown that a circle passing through the origin transforms into a straight line 

 not passing through the origin, whereas any other circle transforms into a circle. If the circle 

 is centered at the origin, so is the transformed circle, but they lie in inverse positions with 

 respect to the unit circle. 



It is sufficient to prove these statements for the inversion. Referring to Figure 27, for 

 P on a given circle C, 



T^ +h^ -2 liT cos 6 = 0?-. 



The result of substituting r = l/r^ where r^ is the value of r at the inverse point Pp may be 

 written .2 i. „2 



2 - 2 » ~ cos 6 = 



1 (A2_a2f "^2-02 (A2_a2)2 



which locates P. similarly on another fixed circle. If A = a = 6 as for 6" in the figure, 

 T = 2h cos €, r^ cos € = 1/(2^) where r^ = 1/r, so that P, is located on the line ST. 



The point 2 = is a singularity to which, in strictness, the transformation does not 

 apply. It is often convenient, however, to speak of a single "point at infinity." If this is 

 done, it can be said that the transformation w = l/z transforms the point 3 = into the point 

 to = oo. If 2 is allowed to approach 2 = in a certain direction, w recedes toward « in a cor- 

 responding direction, and vice versa. If 2 goes around 2 = along a curve of very small 

 diameter, w goes around w = oo along a curve on which | w | is everywhere large, and vice versa. 



The transformation w = I/2 is single-valued in both directions; any point of the 2-plane 

 is transformed into a definite point on the u;-plane, and the inverse transformation 2 = \/w 

 transforms any point on the w-plane into a definite point on the 2-plane. 



Further formulas for this transformation will be found in Section 37. 



The most general transformation that transforms all lines and circles into lines or circles 

 is the bilinear transformation, sometimes called linear, or 



C2 +d 

 where a,b,c, and d are constants, real or complex 



(3). The Transformation 



w = 2^/2 

 on the other hand, is double-valued, transforming every point on the 2-plane except and 

 into two points on the tw-plane. 



44 



