90 



FUNCTIONS OF A COMPLEX VARIABLE. [INT. IV. 



For instance, in the case of the logarithm 

 log z = log r + i<f> + Zmri, 

 u = log r, v = <f> + 2mr, 



as z takes all possible values in the JT F-plane, u = log r varies from 

 oo to + oo but v varies only in limits differing by 27r, so that the 

 whole X F-plane is entirely represented on a strip of the UV- plane 

 infinite in one direction but of the finite width 2?r in the 

 other. This strip is repeated an infinite number of times each 

 giving the same conformal representation of the whole X F-plane. 

 For instance the radii (j> = const, and the circles r = const, cutting 

 them orthogonally in the X F-plane correspond to the lines 

 v = const, u = const, in the CTF-plane. 



FIG. 24. 

 Corresponding regions of the figures are similarly shaded. 



