1550251515-Classical_Complex_Analysis__Gonzalez_

374 Chapter6

Now, using (6.16-13) we get, for J > O,

d7/J J

max( dB)= min lf8(z)l^2

If z 12 - lfz 12

• l = (lfzl - lfa:i)^2 - l

=

2lfzl

lfzl - lfzl

• ( d7/J ) J
  mm dB = max lfMz)l^2

If z 12 - lfa: 12

• l = (lfzl + 1Jzl)^2 - l

= --'-~ -2lfzl

lfzl + IJzl

Case IL If J = 0 and f8(z) =f. O, equation (6.17-2) gives d1f;/dB = -1, so

that 7/J + 0 = C (a constant). In this case 7/J is a strictly decreasing function

of B, and every direction Bis mapped into a fixed direction B' = 7/J + B = C.

When J = 0 the Kasner circle passes through the origin 0, and the

construction in Fig. 6.16 shows that if Bis increased by a, then 7/J decreases

by a, so that

B

1

= ( i/J - OI) + ( B + O'.) = 1/J + B

remains unchanged provided that f8(z) =f. 0.

To determine the value of the constant C we note that for B = 0 we

have 1/J = 7/Ja = ArgOM = Arg(fz + fz) = C (Fig. 6.17). Therefore,

all directions B [except, possibly, those Bo + k7r for which f8(z) = OJ are

mapped into the fixed direction

B' = 7/J + B = 7/Ja .= Arg(fz + fz) (6.17-6)

We have made an exception of the directions Bo + hr because for those

values of B the directional derivative f8(z) vanishes and 7/J = Argf8(z) is

J = 0

Fig. 6.1.6