Residues of complex functions. Hello friends, today I’ll talk about the residues of complex functions. Have a look!!

Want to know more about complex functions?? Do check out:

**Harmonic function and its conjugate function**

**Singularities and zeros of complex functions**

**Residues of complex functions**

(1) The residue of a complex function at a single pole is

(2) The residue of a complex function at a pole of order is

Now I’ll give some examples of residues of complex functions at their poles.

**Solved examples of the residues of complex functions**

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 are the examples.

**Example 1**

According to James et al. (2011)*, “Determine the residues of the following rational function at each pole in the finite plane:

”

**Solution**

Now here the given complex function is: . First of all, I’ll get the poles of the function .

**Step 1**

So I’ll factorise the denominator of to get

Thus the complex function will be

If , then has a singularity at called a pole. Therefore, in this example, the function has simple poles at

Next, I’ll get the residue of the function at its simple pole at .

**Step 2**

As I have mentioned above, the residue of a complex function at a single pole is

Thus the residue of the function at will be

And that means

Then I’ll simplify it to get the residue of the function at as

Next, I’ll get the residue of the function at its simple pole at .

**Step 3**

Again I’ll repeat step 2 for . Thus it will be

And that means

Then I’ll simplify it to get the residue of the function at as

Now I’ll get the residue of at its another simple pole at .

**Step 4**

Again I’ll repeat steps 2 and 3 for . Thus it will be

And that means

Then I’ll simplify it to get the residue of the function at as

Thus the function has three simple poles at . Also, the residues of at these points are and respectively.

Hence I can conclude that this is the solution to the given example.

Now I’ll give you another example.

**Example 2**

According to James et al. (2011)*, “Determine the residues of the following rational function at each pole in the finite plane:

”

**Solution**

Now here the given complex function is: . First of all, I’ll get the poles of the function .

**Step 1**

If , then has a singularity at called a pole. And if , then is a pole of the function of order .

Since, in this example , I can say that it has a simple pole at . Also, is its another pole of order .

Now I’ll get the residues of at its poles. First, I’ll get the residue of at .

**Step 2**

As I have mentioned in example 1, the residue of a complex function at a single pole is

Thus the residue of the function at will be

And that means

Then I’ll simplify it to get the residue of the function at as

Next, I’ll get the residue of the function at its pole at of order .

**Step 3**

As I have mentioned above, the residue of a complex function at a pole of order is

So in this example, the residue of the function at a pole of order will be

And that means

Now I’ll simplify it to get

Next, I’ll substitute the limit to get

So this gives

Thus the summary is: has a simple pole at and its residue is . Also, it has another pole of order at and its residue is also .

Hence I can conclude that this is the solution to the given example.

Now I’ll give you another example.

**Example 3**

According to James et al. (2011)*, “Determine the residues of the following rational function at each pole in the finite plane:

”

**Solution**

Now here the given complex function is: . First of all, I’ll get the poles of the function .

**Step 1**

As I have mentioned in Example 2, if , then is a pole of the function of order . Since in this example , I can say that it has a pole of order at . Now I’ll get the residues of at its pole .

**Step 2**

As I have mentioned above, the residue of a complex function at a pole of order is

So in this example, the residue of the function at a pole of order will be

And that means

Thus I can say that

Now I’ll differentiate it further to get

So that gives

Next, I’ll simplify it. Thus I can say that the residue of the complex function at its pole of order is

Hence I can conclude that this is the solution to the given example.

Dear friends, this is the end of today’s post. 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!!

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