Question

Using the definition (2.1) of show that the following familiar formulas hold.

Short answer

By using the definition 2.1, it is showed that the following familiar formula hold.

Step-by-step solution

Definition 2.1 of analytic function

A function is analytic (or regular or holomorphic or monogenic) in a region of the complex plane if it has a (unique) derivative at every point of the region. The statement is analytic at a point means that has a derivative at every point inside some small circle about .

The derivative of is defined (just as it is for a function of a real variable) by the equation,

where and .

Apply definition 2.1

Given the function

By the definition 2.1,

Solve the function is as follows:

Here it is clear that the following familiar formula hold by using the definition 2.1.