1550251515-Classical_Complex_Analysis__Gonzalez_

Differentiation 355

In particular, if f is monogenic at z, then

( dw d( ) 8 = f' ( z) ( dz d( ) 8

15. Show that

fz =

2

~ 1

2

.rfMz)dB

i.e., the mean derivative at a point equals the integral mean value of

f~(z) on the Kasner circle.

6.12 THE FINITE INCREMENTS FORMULA

If the real function u = u( x, y) is differentiable in some open set A we have
at each point of A the expression

Au = u., Ax + Uy Ay + c1 Ax + c2 Ay (6.12-1)

where c 1 , c2 --+ 0 as Ax, Ay--+ 0 (Section 6.4). This is called the finite
increments formula for real functions of two real variables. From this
formula we can derive a similar result for complex functions of class 1J(A),
as shown in the following theorem.

Theorem 6.21 Let w = f(z) = u(x, y) + iv(x, y) E 'D(A). Let z E A,
z + Az E Nr(z) CA, and Aw= f(z + Az) - f(z). Then we have

Aw = f z Az + fz Az + 'f/1 Az + 'f/2 Az

where 'f/1, 'f/2 --+ 0 as Az --+ 0.

Proof From (6.12-1) and the analogous formula for Av, namely,

Av = v., Ax + Vy Ay + €3 Ax + c4 Ay

we get

(6.12-2)

Aw= Au+i Av= (u., +iv.,) Ax+(uy +ivy) Ay+>.1 Ax+>.2 Ay (6.12-3)

where >. 1 = c1 + ica and >. 2 = c2 + ic4. By using (6.7-11), (6.7-12) and
Ax = %(Az + Az), Ay = -%i(Az - Az), equation (6.12-3) becomes

Aw = f z Az + fz Az + 'f/1 Az + 'f/2 Az

where 'f/l = %(>.1-i>.2), 'f/2 =^1 / 2 (>.1 +i>.2). Clearly, 'f/1, 'f/2--+ 0 as Az--+ 0.
As in real variables, the total differential dw = fz Az+ f-zAZ may be defined
as t~e principal part of the finite increment Aw.