1550251515-Classical_Complex_Analysis__Gonzalez_

328 Chapter^6

and (6.7-7) may be written as df = f'(z)dz. This is the formula for the

differential of a function at a point where it is monogenic.

An analytic function of z in some open set A must be considered as a

function of z alone, i.e., as independent of z. The function f(z) = u +iv is

said to be conjugate analytic in some open set A ifthe function ](z) = u-iv

is analytic in A. Then the Cauchy-Riemann equations at any point of A
are Ux =-Vy and Uy = Vx. In this case formula (6.7-8) becomes

fz = 0

and (6.7-9) gives fz = Ux + ivx = ]'(z).

(6.7-15)

Definitions 6.6 Functions of class 1J for which (6.7-15) holds at a point

are called conjugate monogenic at that point. If (6.7-15) holds in some

open set A where f is defined and of class 1J, then f is called conjugate

analytic (or antianalytic) in A. A conjugate analytic function in A should

be considered as a function of z alone.

Examples For the function f(z) = lzl^2 = zz we have fz = z i= 0 except

at z = 0. Hence this function is nonanalytic in C. However, at z = 0 it is

monogenic. Similarly, fz = z i= 0 except at z = 0. Thus this function is not

conjugate analytic either. At z = 0 it is conjugate monogenic. The function

f(z) = z^2 is conjugate analytic in C, since fz = 0. Clearly, ](z) = z^2 is

analytic in the whole complex plane.

7.7 Elementary Properties of the Complex Integral

OPERATORS Bz AND 8-z

The complex differential operators 8z and 8-z have properties analogous to

those of the partial derivatives of real analysis. It will follow from those

properties that the symbols azf and 8-zf can be interpreted, under certain

conditions on f, as formal partial derivatives off with respect to z and

z, respectively.

By the formal partial derivative with respect to z of a function f(z, z),

defined in some open set A, is meant the following: Suppose that f(z, a),

where a is a constant, has an ordinary derivative with respect to z, and

denote it by fz(z, a). Replacing in this result a by z we obtain a function

fz(z, z) defined in some subset of A. This function will be called the formal

partial derivative off with respect to z. In a similar manner the formal

partial derivative off with respect to z is defined.

In what follows these formal partial derivatives will be denoted 8 f I 8z

and 8f /8z, respectively.