1550251515-Classical_Complex_Analysis__Gonzalez_

320 Chapter^6

y

0 x

Fig. 6,3,

function F(t) = f(z(t)), a::=; t ::=; (3, is differentiable on [a,(J], and

F'(t) = f'(z)z'(t) = 0, a::::; t::::; (3

Hence, F(t) is a constant on [a,(J], i.e., F(a) = F((J), or f(zo) = f(z1).

Definition 6.3 Complex functions f(z) = u +iv with differentiable com-

ponents u and v in some open set A C C will be called functions of class
V(A). The class of analytic or holomorphic functions in A will be denoted
by 1i(A). Since the differentiability of u and v in A implies the continuity

of these functions in A and hence the continuity of f in A, we have

1i(A) c V(A) c C(A)

denoting by C(A) the class of continuous complex functions in A.

6.6 LAPLACE EQUATION. HARMONIC AND

SUBHARMONIC FUNCTIONS

Let w = f(z) = u +iv be defined in some open set A. Suppose that

the partial derivatives ux, uy, Uxy, Vx, Vy, Vxy exist, are continuous in A,

and that the Cauchy-Riemann equations Ux = vy, Uy = -vx are satisfied

in A .. Then, by a well-known theorem of Schwarz, the partial derivatives

Uyx, Vyx also exist and Uxy = Uyx, Vxy = Vyx· Furthermore, from the

Cauchy-Riemann equations it follows that

Uyy = - Vxy, Vyy = Uxy

(6.6-1)

(6.6-2)

so that the second partial derivatives Uxx, Uyy, Vxx, Vyy also exist and are
continuous.
By adding corresponding members of (6.6-1) and (6.6-2) we get

Uxx +uyy = O, Vxx + Vyy = 0