ABSTRACT
Thus the incremental ratio of the function (20.1) has: (i) the same limits along the horizontal (vertical) directions (20.6a) = (20.6b)]:
lim z→0
z−1f (z) =1+ i for y = 0,x→ 0 (20.6a) 1+ i for x= 0,y → 0 (20.6b) 1+ i 2
for x= y → 0 (20.6c) 1+ i 2
for y =−x→ 0, (20.6d)
(ii) also the same values along the diagonals (20.6c) = (20.6d); (iii) the two sets of values are distinct (20.6a) = (20.6b) = (20.6c) = (20.6d). This proves that the function is not holomorphic. This result could be predicted by using (3.17a,b) to write (20.1) as a function of z and z∗:
f
( z+ z∗
2 , z− z∗ 2i
) =
(1+ i) ( 3z+ z∗2
/ z )
4 ≡ g (z,z∗) ; (20.7)
since g(z,z∗) does not depend on z alone, the function cannot be holomorphic (compare with Example 20.4).
