BLACK. — THE NEIGHBORHOOD OP A SINGULAR POINT. 301 



Consider the point 



$ = -1, ^ = 0. 



Let f = fs — 1 and we have 



whence 



^5 = ^^-i'V/r-4?+4. (e) 



2 

 In the same way, about the point 



we have the function 



^-2 



4 = ^ + i'^r-47?^+4 (f) 



In (e) and (f), for the radical we take only that branch which becomes 

 + 2 for zero values of the argument, and for sufficiently small values 

 of $ the functions are analytic when 



Again, consider the point 



f = 2 Vl - »■ , ^ = 1 + 22. 

 Let 



I = ^- + 2 Vl - *■ , ^ = r},+ l-\-2i, 



and we have 



whence 



^7 ^- ^'^^~"~^ +iVl6-16i + ^^^ 4,^,^-8 (1 + 20.?,. (g) 



For the corresponding point 



f=:_2\/l -i, ^ = l + 2^, 



we have the formula 



^8 = ^^^ g '"^^ -iVie - 16t + ^' - 4,782 _ 8(1 + 2i)r,,. (h) 



In (g) and (h), for the radical we take only the branch which becomes 

 + 4 Vl — ^ for zero values of the arguments, the same value of the 

 radical \/l — i being taken in all cases. These functions are analytic 

 for sufficiently small values of ^ when 



1^7 I = 1^8 I < 2 — C7 . 



Also, considering the corresponding points of 



