A FEW EXAMPLES IN THE THEORY OF FUNCTIONS 245 



as indicated, w 2 and w 3 in the w-plane move in such a manner that 

 iv 2 changes to w 3 and w 3 into w 2 . Those roots which at one of the points 



2 

 ± T7 = become e( l ual are permuted if z describes a complete circle 



around this one point. From (2) it is apparent that when z moves 



around — = the roots w 1 and w 2 in (2) are permuted, while when z 



v 27 



2 

 moves around — —= the roots w 2 and w 3 are permuted. 



Designating these points by B x and B 2 , we may let z describe a path 

 passing through one of these points, say B xt around which the roots w x 

 and w 2 are permuted. These two points w x and w 2 describe paths in the 



w-plane which pass through the point \- in the w-plane corresponding 



to B t in the z-plane. But as for any point z, w can have either value w x 

 or w 2 , it is clear that each of the points in the w-plane representing w x 

 and w 2 can move on either of the two branches. Such a point B x , and 

 similarly B 2 , is called a branch- point of the function w. 



(c) Construction of the Paths Described by z and w. 



To trace the curves which w describes when z describes given curves, 

 write z=x+iy and w=u + iv, so that the original equation becomes 



(u+ivY — (u+iv)+x+iy=o . 

 Separating real and imaginary parts, we get 



m 3 — $uv 2 — u=— x , (2) 



3U 2 v—v 3 —v=—y . (4) 



It now z describes the curve }(x, y)=o, then w describes the curve 



/[(— U3+3UV 2 +U) , (—3U 2 V+V* +z;)l= . 



If the curve in the z-plane is x=+— 2= (v = arbitrary), then w in 



V 27 



the w-plane moves on the curve 



u^ — t ) uv 2 -u-\ — t==o , CO 



1/27 V3/ 



a cubic which has a ^double-point at u = \- , corresponding to the 



