A FEW EXAMPLES IN THE THEORY OF FUNCTIONS 25 1 



From the Riemann surface it appears that when z turns three times 

 about both branch-points the function will return with its original 

 determination. Such a path of 2, in this example, is also equivalent 

 with a path about the point 00 , and around this point the three deter- 

 minations of the function are permuted. To prove this analytically, 

 we expand the function in the neighborhood of 2=00 ; or z'=o, when 



z f =- . w becomes infinite when 2=00 , but -tj= is finite for 2=00 . 



2 fz 



Indeed, 



Km 



Z=oO 



(ry*=^z-^-^i-<- V'-£)|- 



IV 



Thus, it is possible to expand the function w^ l) =-jr= about 2=00 , or 



V z 



iu I =iv.z'z about 2' = o. Substituting w=w 1 .z f ~* in w 3 —w-\-z = o, we 



get w^—w^z'* + 1=0. If z' turns three times about the origin, w x 



and hence w will return with its original determination. Hence, 



putting z , = z 1 3 , w will return with its original determination when z t 



turns once about the origin and consequently w x in the equation 



IVi 3 — w 1 .z 1 2 + i=o 



can be expanded like a uniform function. Writing w 1 = f(z I ) we have 



Wl =/(o)+ 2l ./'(o)+^/"(o)+... . 



2Z ID 



Now /(o)=— 1, and from w l 3 —w 1 .z 1 i + 1=0 we find w T f = 



Forz x = o, w 1 =—i, andw I '=o(f(o) = o) . 



6lVj 2 + 2W 1 Z l 2 —6w l 2 Wi 'Zi — 2Z 1 3 Wi 



**x"=/"(*x) ! 



(3W 1 2 -2Z l 2 y 



2 



which for z T = o becomes - . Hence 



Wj=-H — z^-f-. . . , or 

 3 



; I = -i+-z , § + . .. , 

 3 



or 



w l = — I+-Z - § + . . . . 



3 



Here w t does, of course, not mean the first determination. 



