1550251515-Classical_Complex_Analysis__Gonzalez_

346 Chapter6

Next suppose that

Then we have

or

lfzl^2 + l!zl^2 + fzfze^2 i^8 + fzf-ze-^2 i^8 = C^2

Letting il = fzfz, B = lfzl^2 + l/zl^2 - C^2 , ( = e^2 i^8 , the equation above

becomes

A(^2 +B(+A = o

The preceding equation must be satisfied for infinitely many values of (
(those on the unit circle). Hence

A= fzfz = 0 and

so that either f z =. 0 or f-z = 0.

Theorem 6.18 Suppose that at a fixed point z we have JO(z) I- 0 for all

0(0 ~ 0::;; 27r). Then Arg/O(z) = C (a constant) iff f is monogenic at z.

Proof If fz = 0, we have JO(z) = fz I-0, so that Argf~(z) = Argfz =a
constant at z. Now assume that Arg/~(z) = ,,P = a constant (i.e.,
independent of 0). From

we obtain

and

or

f~(z) = pei..P = e-i1J(D9u + iD9v)

ei(.,P+IJ) = ~ (D9u + iD9v)

' p

D9v

tan( ,,P + 0) = -D

!JU

D9v

,,P = Arc tan -D - 0 + k7r

!JU

for a suitable integer k. Hence by taking derivatives with respect to 0,
we find that

0

= d,,P = (Deu)(Dev)' -(Dev)(Deu)' _

1

dO (Deu)^2 + (Dev)^2

J

= -p2 -1 (6.10-10)