dz 



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^(6/a) 



Since dw^dz does not vary from point to point of the s-plane, even finite figures will transform 

 conformally under this transformation; their linear dimensions, however, will be changed on the 

 to-plane in the ratio yja^ + b'^ and they will be rotated, relatively to the real axis, through an 

 angle equal to amp A or to tan~^(6/(2). They will also be displaced in the direction of the 

 vector representing B. 



(2). Another interesting transformation is the inverse transformation 



,„ _ 1 



_ 1 



The transformation from the 2-plane to the w-plane may be imagined to be made in two steps. 

 First, let each point P at r distance from the origin of z be moved to a position P' lying on the 

 same radius from the origin but at a distance 1/r; that is, each point is displaced to its inverse 

 point in the unit circle, r = 1. Such a geometrical transformation is called inversion with re- 

 spect to the circle. It can be visualized by imagining the plane to be turned inside out while 

 the unit circle stands still. Then let each point be moved to its mirror image in the real axis; 

 this changes the sign of d. Thus the inverse transformation is equivalent geometrically to 

 inversion in the unit circle plus a reflection in the real axis. These two steps may be taken 

 in either order. The changes may be imagined to be' executed on the 3-plane, which is then 

 rechristened the w)-plane; see Figures 25 and 26. 



Figure 25 - The transformation w = l/z Figure 26 - The transformation w = 1/z. 



43 



