1550251515-Classical_Complex_Analysis__Gonzalez_

372 Chapter^6

We distinguish the following cases:

Case I. If J > 0, the maximum and minimum values of ~b occur when

J = lf~(z)l^2 or Vi= lf~(z)I (6.17-3)

This result has an obvious geometric interpretation. In the case J > 0

the coordinate origin 0 is exterior to the Kasner circle, JJ represents the

length of the segment of tangent from the origin to the circle (see Exercises

6.2, problem 9), and (6.17-3) shows that the extreme values of~ namely,

~i and ~ 2 , occur for values of (} for which the corresponding f~(z) are

located at the points of contact Ti and T 2 of the circle with the tangents

issued from 0 (Fig. 6.15). ,

These values of 0 are given by (6.17-3), or

E cos^2 0 + 2F cos 0 sin 0 + G sin^2 0 = J

which may be written as

(G - J)tan^2 0 + 2Ftan0 + (E - J) = 0

The values Oi and 02 corresponding to the points Ti and T 2 , respectively,

can also be determined from Fig. 6.15. In fact, if we let w = Arg fz,

w' = Arg fz, and a = LOCTi, we have

w^1 - 20i = w - 7r + a

so that

-20

Fig. 6.15