analytic harmonic
and
functions
Overview
Does the notion of a derivative of a complex function make sense? If so, how should it be defined and what does it represent? These and similar questions are the focus of this chapter. As you might guess, complex derivatives have a meaningful definition, and many of the standard derivative theorems from calculus (such as the product rule and chain rule) carryover. There are also some interesting applications. But not everything is symmetric. You will learn in this chapter that the mean value theorem for derivatives does not extend to complex functions. In later chapters you will see that differentiable complex functions are, in some sense. much more "differentiable" than differentiable real functions.
3.1
DIFFERENTIABLE FUNCTIONS
AND ANALYTIC
Using our imagination, we take our lead from elementary calculus and define the derivative of f at zo, written J' (zo), by
f / (zo)
=
lim
Z--->ZQ
f (z) - f (zo) ,
Z Zo
provided the limit exists. If it does, we say that the function f is differentiable at zoo If we write 6.z = z - zo, then we can express Equation (3-1) in the form
f
I (
Zo -
)
_
Ll.z--->o
l' 1m
f (zo + 6.z) - f (zo)
uZ
A •
If we let w = 1(z) and 6.w = notation ~~ for the derivative:
1(z) - 1(zo),
then we can use the Leibniz
1 (zo) = -dz =
,
dw
6.w
L>Z--->O DoZ
lim ~.
I' (zo) =
lim -z--->zo
Z3 -
z5
(z2+zoz+z5) z - Zo
z - Zo
= hm --------z--->zo
Z-+ZO
.
(z-zo)
= lim (z2 + ZOZ + z5) = 3z5·
Pay careful attention to the of the limit must be independent two curves that end at Zo along does not have a limit as. 6.z ---+ same observation applies to the • EXAMPLE differentiable. 3.2
complex value 6.z in Equation (3-3); the value of the manner in which 6.z ---+ O. If we can find which approaches distinct values, then 0 and 1 does not have a derivative at zo0 The limits in Equations (3-2) and (3-1) .
1~
1~
Show that the function w =
1(z)
= z =
x - iy is nowhere
We choose two...