BLACK. — THE NEIGHBORHOOD OP A SINGULAR POINT. 301 

 Consider the point 



Let £ = £ 5 — 1 and we have 



& 2 - 2 & + ?-&£+ £=0, 



whence 



^ = ^- 2 -i^-^+4. 



(e) 



In the same way, about the point 



£=1, ^ = 0, 



we liave the function 



£ 



6= — + W£ 2 -4?+4 (f) 



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

 -f 2 for zero values of the argument, and for sufficiently small values 

 of f the functions are analytic when 



| rj | < 1 — e 2 . 



Agaiu, consider the point 



Let 



1 = & + 2 Vl - i , ^ = -77 + 1 + 2*, 



and we have 



^ 7 2 + 4 yi^^7 4- 7 + Vl 2 + 2 (1 + 2 1) 77 7 - £ 7 £ - 2 vT^?£ = 0, 

 whence 



j^- 4 ^ 1 -*-^ 1^16-16^+^-4^-8 (l + 2^ 7 . (g) 



For the corresponding point 



4 r =-2A/l -i, ^ = l + 2», 



we have the formula 



4 Vl — * + £ 



& = 2 ^ - £ Vl6 - 16* + ? - 4 V - 8(1 + 2t) % . (h) 



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

 + 4 V4 — i 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 



I *77 | = 1 17s | < 2 — e 7 • 

 Also, considering the corresponding points of 



