Singularities and zeros of the complex numbers. Hello friends, today I’ll talk about the singularities and zeros of the complex numbers. Have a look!!

**Singularities and zeros of the complex numbers**

(1). The singularity of a complex function is a point in the plane where ceases to be analytic.

(2). A zero of is a point in the plane at which .

(3). If , then has a singularity at called a pole.

(4). If , then is a pole of the function of order .

(5). If , where for , then is the essential singularity of .

Now I’ll give some examples of singularities and zeros of the complex numbers.

**Solved examples on singularities and zeros of the complex numbers**

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 location of, and classify the singularities and zeros of the following functions. Specify also any zeros that may exist. .”

**Solution**

So let’s say that the complex number is

As I have mentioned above, a zero of is a point in the plane at which .

Now here the numerator is . So it can never be zero. Also, the denominator is only infinite when is infinite. Thus has no zeros in the finite plane.

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

Thus the denominator is when

So this means either or .

(1)

Next, I’ll get the value of for which . Since I can’t factorise it, I’ll use the standard formula to get the value of . Therefore it will be

Then I’ll simplify it to get

(2)

Therefore from equations (1) and (2), I can say that the singularities are at

Thus the simple poles are at . And the pole of order is at .

Hence I can conclude that these are the answers to the given example.

Now I’ll give you another example.

**Example 2**

According to James et al. (2011)*, “Determine the location of, and classify the singularities and zeros of the following functions. Specify also any zeros that may exist. . ”

**Solution**

So let’s say that the complex number is

As I have mentioned above, a zero of is a point in the plane at which .

Therefore for ,

Now this gives

Also, will be , if the denominator is infinity. So, just like example 1. the denominator is only infinite when is infinite. And thus has a single zero in the finite plane at .

Also, if , then has a singularity at called a pole. Thus the denominator is when

So this means

So the singularities are at

Out of these, a single zero is at . Also, are simple poles.

Hence I can conclude that these are the solutions to the given example.

Now I’ll give you another example.

**Example 3**

According to James et al. (2011)*, “ Determine the location of, and classify the singularities and zeros of the following functions. Specify also any zeros that may exist. . ”

**Solution**

So let’s say that the complex number is

As I have mentioned above, a zero of is a point in the plane at which .

Therefore for ,

Also, will be , if the denominator is infinity. So, just like examples 1 and 2. the denominator is only infinite when is infinite. And thus has a single zero in the finite plane at .

Again the denominator is when

Now I’ll factorize it to get the values of . So it will be

Thus the values of are

So are all simple poles.

Therefore I can say that a single zero is at and are all simple poles.

Hence I can conclude that these are the solutions to the given example.

Now I’ll give you another example.

**Example 4**

According to James et al. (2011)*, “ Determine the location of, and classify the singularities and zeros of the following functions. Specify also any zeros that may exist. . ”

**Solution**

So let’s say that the complex number is

As I have mentioned above, if , where for , then is the essential singularity of .

If is infinity for , then the function has an essential singularity at .

Now will be infinite if

So this gives

Thus is the essentially singular point.

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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