A FEW EXAMPLES IN THE THEORY OF FUNCTIONS 



243 



It is important to observe that the expressions (1), in the same order, 

 do not correspond to the expressions in (2). For instance, for 2=0; i. e., 



<f> = - , the values in (2) become 



w x = +i , 

 w 2 = o , 

 w 3 = -i . 



The seeming contradiction can be explained by the fact, that in (2) 

 the cube-roots are explicitly extracted, while 

 in (1) p and q simply indicate cube-roots and 

 each may just as well be replaced by either 

 ap or or a 2 p and a 2 q or aq y respectively. For- 

 mulas (2) are therefore the proper determi- 

 nations of the function w within the region 

 2 ^2 



--=<2<- 7 = 

 I/27 I/27 



i, e., <f> = 7r, 



2 

 (z real). When z= —7= 

 T/27 



W-, = —2* 



When z= ——7= 

 V 27 



i. e., <f> = o, 



w— J; , 



W,=Wf. 



Fig. 



2 

 Hence, for z=+—j= , the roots w x and w 2 become equal, while 

 V 27 

 2 

 for z= — 7= the roots w 2 and w, become equal. 

 v 27 



