1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

cba~yp.t~l 3

harmonic 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 cal-
culus (such as the product rule and chain rule) carry over. There are also some
interesting applications. But not everything is symmetric. You will learn in
this chapter that the mean value theorem for derJvatives does not extend to
complex functions. In later chapters you will see that differentiable complex
functions are, in some sense, much more "differe ntiable" than differentiable
real functions.

3.1 Differentiable and Analytic Functions

Using our imagination, we take our lead from elementary calculus and define the

derivative off at zo, written f' (zo), by

f

, ( )

1

. f (z) - f (zo)
zo= 1m ,
z-zo Z - Z o

(3-1)

provided the limit exists. If it does, we say that the function f is differentia ble

at z 0. U we write 6.z = z -zo, then we can express Equation (3-1) in the form

!

'( )

1

. f(zo+l:!.z)-f(zo)
ZO = !ID A •
llz-o u z

(3-2)

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