1550251515-Classical_Complex_Analysis__Gonzalez_

Differentiation 395

The following assumptions are made concerning the coefficients O'k, 7k:

(1) They are defined for all values of x and y, respectively; (2) they are

analytic functions of their respective real variables; (3) they are positive;

and (4) their limits as x ~ ±oo, or as y ~ ±oo, exist and are =f=. O, oo.

The Laplace-type second-order equations obtained from (6.24-9) by

eliminating either v or u are

( O'l Ux) + ( 0'

(^2) Uy) = Q
71 x 72 y
and
(
7
2 Vx) + (
7
1 Vy) = Q
0'2 x 0'1 y

The generalized derivative off, denoted J', is defined by

..

i .. i

J'(z) = 0'1Ux + -Vx = 71Vy - -Uy

0'2 72

The integral of a continuous function along a rectifiable arc is then defined

and analogous to a number of theorems in analytic function theory are

derived.

(e) S. Bergman in [15] and L. Bers in [8] and [9] consider a generalized

Cauchy-Riemann system of the form

Ux = u(x,y)vy, Uy= -u(x,y)vx (6.24-10)

where u(x,y) is defined and bounded in a domain D, O' > 0, and O'x, O'y

exist and satisfy in D a Holder condition.

According to Bers, a complex function f = u +iv is pseudoanalytic with

respect to a (or a a-analytic function) at a point z 0 if in some neighborhood
of this point the partial derivatives of u and v exist, are continuous, and
satisfy (6.24-10). A function is pseudoanalytic in a domain if it is pseudoan-
alytic at every point of the domain. A constant function is a trivial example
of a pseudoanalytic function (with respect to any a). An interesting result
is that pseudoanalytic functions also define open mappings..
(f) In his book [10] (see also (11]), Bers generalizes analytic function
theory by considering two functions F( z) and G( z) defined in a domain

Do and subjected to the following conditions: (1) Im[F(z)G(z)] > 0 in D 0 ,

and (2) Fz, F 2 , Gz, G 2 exist and are Holder-continuous in Do. It follows

that for every Zo E D there are unique real constants >.o, μo such that

w(zo) = >.oF(zo) + μoG(zo)

Thus the functions F and G, called a generating pair in Do, play the role
of 1 and i in ordinary function theory.