Differentiation 371J>O J<O J = 0Fig. 6.14By considering the Kasner circle of f at z it is easy to see that the
maximum and minimum values of p correspond to those values of B for
which the point ( = f M z) lies at the endpoints of the diameter of the circlethat contains the origin 0. There are two values of ((differing by rr) for
each extremum, since the Kasner circle is described twice by ( as B variesfrom 0 to 2rr. In Fig. 6.14 the three cases J > 0, J < O, and J = 0 are
illustrated (assuming that lfz I + If z I -::f 0).
Since OC = lfzl and the radius of the Kasner circle is given by lfzl,
it follows that:
l. If J > 0, then Pl = lfzl + lfz:I, P2 = lfzl -lfzl·
- If J < 0, then Pl = lfzl + lf.zl, P2 = lfzl -lfzl·
- It J = 0, then Pl = lfzl + lfzl, P2 = 0.
Thus in any case,P1 = lfzl + lfzl and P2 = llfzl -lfzll
6.17 NONCONFORMAL MAPPINGS. THE ANGULAR
DISTORTION t/J AS A FUNCTION OF 9
(6.16-13)Assuming that f~(z) -::f 0 and letting 'ljJ = Argf 0 (z) in Section 6.10,
Theorem 6.18, we have obtained the formula
D 6 v ·
'ljJ =Arc tan - - B + k7r (6.17-1)
Deufrom which we derived
d'ljJ J
df) = lf6(z)l^2 -
1 (6.17-2)