1550251515-Classical_Complex_Analysis__Gonzalez_

Differentiation 321

Hence, under the foregoing assumptions, the components u and v of an

analytic function on an open set A satisfy at every point of A the Laplace

equation in two dimensions, namely,

a2'1/J a2'lfJ

b..'l/J = 8x2 + 8y2 = 0 (6.6-3)

This equation occurs frequently in mathematical physics. For instance, it is

satisfied by the velocity potential and stream function of a two-dimensional

irrotational flow of an incompressible nonviscous fluid.

Definition 6.4 Real-valued functions of two real variables 'l/J: A -+ JR.

of class c<^2 >(A), i.e., with continuous partial derivatives on A up to the
second order at least, and satisfying equation (6.6-3) are called harmonic
functions or potential functions in A. If 'l/Jxx + 'l/Jyy 2:: 0 in A, then 'l/J is
called subharmonic.
By the-Laplacian of 'l/J is meant the sum 'l/Jxx + 'l/Jyy, denoted b..'l/J and
also \7^2 1/J. The two-dimensional Laplacian differential operator b.. (or \7^2 )
is defined in Cartesian coordinates by

a2 a2

b.. = 8x^2 + 8y^2

and a similar definition is adopted in higher-dimensional Euclidean spaces.

For instance, in the case n = 3 we write

a2 a2 a2

!}._ = 8x^2 + 8y^2 + 8z^2

x, y, z being the current Cartesian coordinates.

If D C JR.^2 is a domain, and 'l/J: D -+ JR., we say that 'l/J is harmonic at

(x 0 , y 0 ) E D if there is some open set A such that (xo, Yo) E A C D and

such that 'ljJ I A is harmonic in A. A complex function f: A-+ C, where

f = u +iv, is called harmonic in A if both u and v are harmonic functions

in A. Similarly, f is harmonic at a point z 0 = ( x 0 , y 0 ) in A if both u and

v are harmonic at ( x 0 , y 0 ). The assumption that all partial derivatives of

the second order of 'l/J exist and are continuous implies the continuity of 'l/Jx

and 'l/Jy, as well as that of 'l/J.

In Section 7.21 we prove that an analytic function in an open set A has

derivatives of all orders in A. This implies the continuity of all those deriva-

tives, as well as the existence and continuity of all partial derivatives of the

components u and v. Hence the assumption made at the beginning of this

section concerning the existence and continuity of the partial derivatives

Uxy and Vxy is automatically satisfied for an analytic function. It follows

that the components u and v of an analytic function in an open set A are

always harmonic functions in A, so f itself is harmonic in A. However, not