which of the following function is analytic
a) f (Z)=son Z
b) f (Z)=Z
c)f (Z)= Im (Z)
Answers
Answer:
The main goal of this topic is to define and give some of the important properties of
complex analytic functions. A function f(z) is analytic if it has a complex derivative f
0
(z).
In general, the rules for computing derivatives will be familiar to you from single variable
calculus. However, a much richer set of conclusions can be drawn about a complex analytic
function than is generally true about real differentiable functions.
2.2 The derivative: preliminaries
In calculus we defined the derivative as a limit. In complex analysis we will do the same.
f
0
(z) = lim
∆z→0
∆f
∆z
= lim
∆z→0
f(z + ∆z) − f(z)
∆z
.
Before giving the derivative our full attention we are going to have to spend some time
exploring and understanding limits. To motivate this we’ll first look at two simple examples
– one positive and one negative.
Example 2.1. Find the derivative of f(z) = z
2
.
Solution: We compute using the definition of the derivative as a limit.
lim
∆z→0
(z + ∆z)
2 − z
2
∆z
= lim
∆z→0
z
2 + 2z∆z + (∆z)
2 − z
2
∆z
= lim
∆z→0
2z + ∆z = 2z.
Step-by-step explanation: