Complex Analysis - Maths KU

142 Chapter 3 Analytic Functions
3.1 Differentiability and Analyticity
The calculus of3.1 complex functions deals with the usual concepts of derivatives and integrals
of these functions. In this section we shall give the limit definition of the derivative of a
complex function f(z). Although many of the concepts in this section will seem familiar to
you, such as the product, quotient, and chain rules of differentiation, there are important
differences between this material and the calculus of real functions f(x). As the subsequent
chapters of this text unfold, you will see that except for familiarity of names and definitions,
there is little similarity between the interpretations of quantities such as f ′ (x) and f ′ (z).
The Derivative Suppose z = x+iy and z0 = x0+iy0; then the change
in z0 is the difference ∆z = z − z0 or ∆z = x − x0 + i(y − y0) =∆x + i∆y. If
a complex function w = f(z) is defined at z and z0, then the corresponding
change in the function is the difference ∆w = f(z0 +∆z) − f(z0). The
derivative of the function f is defined in terms of a limit of the difference
quotient ∆w/∆z as ∆z → 0.
Definition 3.1 Derivative of Complex Function
Suppose the complex function f is defined in a neighborhood of a point
z0. The derivative of f at z0, denoted by f ′ (z0), is
provided this limit exists.
f ′ f(z0 +∆z) − f(z0)
(z0) = lim
∆z→0 ∆z
If the limit in (1) exists, then the function f is said to be differentiable
at z0. Two other symbols denoting the derivative of w = f(z) are w ′ and
dw/dz. If the latter notation is used, then the value of a derivative at a
specified point z0 is written dw
�
�
� .
dz
� z=z0
EXAMPLE 1 Using Definition 3.1
Use Definition 3.1 to find the derivative of f(z) =z 2 − 5z.
Solution Because we are going to compute the derivative of f at any point,
we replace z0 in (1) by the symbol z. First,
f(z +∆z) =(z +∆z) 2 − 5(z +∆z) =z 2 +2z∆z +(∆z) 2 − 5z − 5∆z.
Second,
f(z +∆z) − f(z) =z 2 +2z∆z +(∆z) 2 − 5z − 5∆z − (z 2 − 5z)
=2z∆z +(∆z) 2 − 5∆z.
(1)