1550251515-Classical_Complex_Analysis__Gonzalez_

Differentiation 319

As ~z ~ 0 the last two terms in (6.5-7) tend to zero. Therefore,

f'( z ) = tl.z-+0 Ii Ill ~ ~w uz = u., + zv.,.

i.e., the derivative off exists at z and has the value u., +iv.,.

Example Let w = ez = e"' (cosy + i sin y ). In this case we have

so that

u = e"' cosy,

u., = e"' cosy,

Uy= - e"' siny,

v = e"' siny

Vx =ex siny

Vy= e"' cosy

Since the partial derivatives u., and Uy exist and are continuous for all

(x, y) E JR^2 , u is differentiable in JR^2 ; similarly, vis differentiable everywhere

in JR^2 • In addition, we have u., =Vy and Uy = -v.,. Hence, by Theorem 6.5,

the function w = ez has a derivative for every z E C, and

dw dz = e x cosy + ie · x sm • y = e z

Theorem 6.6 If f is analytic in a region R and f' ( z) = 0 for all z E R,

then f(z) is a constant function in R.

Proof By (6.5-4), f'(z) = 0 for all z E R implies that

Ux = Uy = Vx = Vy = 0
throughout R. Let zo = ( xo, Yo) and z1 = ( x1, Y1) be any two distinct points
of R (Fig. 6.3). Since R is connected, the points (xo, y 0 ) and (xi, Y1) can be
joined by a polygonal line with each side parallel to a coordinate axis (Exer-
cises 2.5, problem 8). By the corresponding theorem for real differentiable
functions of a real variable over some interval, the functions u( x' y) and
v(x, y) are constants along every horizontal or vertical line segment in R,
and thus they are constants along every polygonal line with sides parallel to
the coordinate axes. Hence u(xo,Yo) = u(xi,y1) and v(xo,Yo) ~ v(x1,Y1),
so that f(zo) = f(z1). Since zo and z1 are arbitrary points of R, the
theorem follows.

Remark From u., = 0 in R it does not follow necessarily that u is inde-

pendent of x in R (see [42], p. 121). However; if both u., = 0 and Uy = 0

in a region R, then u is a constant in R, i.e., u is independent of x and y

in that region, as we have shown above.

An alternative proof of.Theorem,6.6 without use o{ (6.5-4) can be given

as follows: Let z 0 and z 1 be any two points of R, and let 1: z = z(t),

a :5 t :5 /3, be a regular arc connecting zo and z1. Then the composite