Functions of complex variables. Hello friends, today it’s about the functions of complex variables. Have a look!!

**Functions of complex variables**

If you’re looking for more in complex numbers, do check in:

**Addition and subtraction of complex numbers**

**Multiplication and division of complex numbers**

**Polar form of a complex number**

**Modulus and argument of a complex number**

functions of complex numbers_compressed

Let’s suppose is a complex number. And is a function. Also, is a variable in the function .

To make myself clear, I just give an example. Suppose the function is

Now here are constants.

Then this function is called the function of a complex variable .

Now I’ll show some examples of that.

**Examples of functions of complex variables**

Disclaimer: None of these examples are mine. I have chosen these from some book or books. I have also given the due reference at the end of the post.

So here is the first example of the functions of complex variables.

**Example 1**

According to Kreyszig (2005)*, “Find Re and Im . Also find their values at the given point . .”

**Solution**

Now here I have the given function is . Let’s choose as the complex variable of the function .

And my task is to find out the real and imaginary parts of .

First of all, I’ll substitute in the function .

**Step 1**

Thus it will be

Next, I’ll expand it to get

Now I’ll simplify it to get

Since , I can say

Hence I can separate the real and imaginary parts of the function as

So I can say that Re is

Also, Im is

Now I’ll find out the values of Re and Im at .

**Step 2**

If I compare with , I can say and .

Thus the real part of the function at will be

So this gives

Similarly, the imaginary part of the function at will be

So this gives

Hence I can conclude that and at are the answers to this example.

Now I’ll give another example of the functions of complex variables.

**Example 2**

According to Kreyszig (2005)*, “Find Re and Im . Also find their values at the given point . .”

**Solution**

Now here I have the given function is . Let’s choose is the complex variable of the function .

And my task is to find out the real and imaginary parts of .

First of all, I’ll substitute in the function .

**Step 1**

Thus it will be

Next, I’ll expand it to get

Since , I can say

Now I’ll remove the imaginary part of the denominator.

Thus it will be

Next, I’ll simplify it to get

Since , I can say

So this gives

Hence I can separate the real and imaginary parts of the function as

So I can say that Re is

Also, Im is

Now I’ll find out the values of Re and Im at .

**Step 2**

If I compare with , I can say and .

Thus the real part of the function at will be

So this gives

Similarly, the imaginary part of the function at will be

So this gives

Hence I can conclude that and at are the answers to this example.

Dear friends, this is the end of today’s post on the functions of complex variables. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

Maritime universities in Chennai says

Great article about the FUNCTIONS OF COMPLEX VARIABLES