u 



Figure 27 — Circles go into circles under the transformation w = 1/z. 



In terms of z = re '", the two values of w are 



to, = 2^/^= r 



1/2 rcos/-i d\ + i sinf 1 dU 



= 2^/2 =-ri/2 



As 2 moves about on its plane, w. and w both move about on the tc-plane; their values 

 are said to constitute different branches of the function z^^'^. The relationship is not like that 

 of the branches of a tree, however, but rather like that of the various loops of a string tied in 

 an open knot. 



To study the situation more closely, let z start from the positive real axis and explore 

 the 2-plane without ever passing directly from the negative real axis to points below it or vice 

 versa; it may move along curves such as ab, ac in Figure 28. The s-plane may be thought of 

 as cut apart just below tlie negative real axis. Let 6 be defined so that -n <. 6 ^ n. Then 

 w. will explore the right-hand half of tlie w-plane, including the positive half of the imaginary 

 axis, wiiile w- explores the other half of the plane. In this way to. maps the entire 2-plane 



45 



