1550251515-Classical_Complex_Analysis__Gonzalez_

356 Chapter 6

It should be noted that a·t each point z the linear function T: C -t C

defined by

T(~z) = fz~z + fz~z

is precisely the differential off in the Frechet's sense. In fact,·from (6.12-2)
we get

1

. lf(z + ~z) - f(z) -T(~z)I
1
. l771~z + 7]2~zl
lffi = lffi
az,.o l~zl az,.o l~zl

= az__,.O lira I 771 + (^772) t..:>.Z ~z I = 0
Since
T(l) = fz + fz = u., +iv.,
and
T(i) = i(fz - fz) =Uy+ ivy
the matrix of the transformation T is the transpose of [ u., v.,] , namely,
Uy Vy
the Jacobian matrix [ u., Uy].
Vx Vy
The values of fz and fz at z may simply be thought as the coefficients
of ~z and ~z, respectively, in the expression of the Frechet differential in
the complex coordinate system (z, z).

6.13 EXISTENCE OF A LOCAL SINGLE-VALUED

INVERSE FUNCTION TO w = /(z)

The following theorem holds:

Theorem 6.22 Let w = f(z) E '.D(A), A open, and suppose that lfzl "f=.
lfzl in some nonempty open subset A' CA. Let z 0 EA', B' = f(A'). Then
Wo = f(zo) E B' has in a sufficiently small neighborhood N 0 (zo) just one
inverse :image, namely, z 0 [50].

Proof Let zo + ~z E Nr(zo) C A' and ~w = f(zo + ~z) - f(z 0 ). From
formula (6.12-2) we obtain

l~wl ~ lfz .6z + fz ~zl - 1771 ~z + 772 ~zl

~ llfzl-lfzlll~zl -(17711 + l772Dl~zJ

= {Jlfzl - JfzJI -(771J + J7721)} J~zJ