Analytical descriptions of functions in Fourier series. Hello friends, today I’ll talk about the analytical descriptions of functions in Fourier series.

Have a look!!

### Analytical descriptions of functions in Fourier series

#### Solved examples of analytical descriptions of functions in Fourier series

Note: None of these examples is mine. I have chosen these from some books. I have also given the due reference at the end of the post.

So here is the first example.

##### Example 1

According to Stroud and Booth (2011), “For the following graph give the analytical description of the function drawn.

##### Solution

Now I have to find out the analytical description of this graph.

So this means that I have to describe the function .

Hence comes my first step.

###### Step 1

As I can see from the graph, the function is discontinuous at . And it repeats itself, first, at and then at .

So I can say that the function has a period .

(1)

Now I’ll describe the function.

###### Step 2

As I can see from the graph, at , the value of is also . So one point is .

And at , the value of is . So I’ve got the other point as .

Now I’ll find out the equation of the straight line passing through two points and .

As we all know, the standard equation of a straight line passing through two points and is

So the equation of the straight line passing through two points and will be

Now I’ll simplify it to get

So this means

Next, I’ll replace with . Thus the function becomes when is in between and .

In mathematical term, I can say

(2)

Now I’ll find out the expression for when is in between and .

###### Step 3

As I can see from the graph, the value of is constant at when is in between and .

(3)

Therefore I can combine equations (1), (2) and (3) to describe the function as

Hence I can conclude that this is the answer to the above-mentioned example.

Now I’ll give another example.

##### Example 2

“For the following graph give the analytical description of the function drawn.

##### Solution

Now here again I’ll find out the analytical description of this graph.

So again I’ll describe the function .

Hence comes my first step.

###### Step 1

As I can see from the graph, the function is discontinuous at and . And it repeats itself at .

So I can say that the function has a period of .

(4)

Now I’ll describe the function.

As I can see from the graph, the value of is constant at when is in between and .

(5)

Now I’ll find out the expression for when is in between and .

###### Step 2

As I can see from the graph, at , the value of is . So one point is .

And at , the value of is . So I’ve got the other point as .

Now I’ll find out the equation of the straight line passing through two points and .

As we all know, the standard equation of a straight line passing through two points and is

So the equation of the straight line passing through two points and will be

Now I’ll simplify it to get

So this means which gives .

Next, I’ll replace with . Thus the function becomes when is in between and .

In mathematical term, I can say

(6)

Now I’ll find out the expression for when is in between and .

###### Step 3

As I can see from the graph, at , the value of is . So one point is .

And at , the value of is . So I’ve got the other point as .

Now I’ll find out the equation of the straight line passing through two points and .

So the equation of the straight line passing through two points and will be

Now I’ll simplify it to get

So this means which gives .

Next, I’ll replace with . Thus the function becomes when is in between and .

In mathematical term, I can say

(7)

Therefore I can combine equations (4), (5), (6) and (7) to describe the function as

Hence I can conclude that this is the answer to the above-mentioned example.

Now I’ll give another example.

##### Example 3

“For the following graph give the analytical description of the function drawn.

##### Solution

Now here again I’ll find out the analytical description of this graph.

So again I’ll describe the function .

Hence comes my first step.

###### Step 1

As I can see from the graph, the function is discontinuous at .

Also, I can see that the function starts with .

So I’ll start when is in between and .

At , the value of is 1. So one point is .

And at , the value of is . So I’ve got the other point as .

Now I’ll find out the equation of the straight line passing through two points and .

So the equation of the straight line passing through two points and will be

Now I’ll simplify it to get

So this means which gives .

Next, I’ll replace with . Thus the function becomes when is in between and .

In mathematical term, I can say

(8)

Next, I’ll choose the interval between and .

###### Step 2

As I can see from the graph, the value of is constant as when is in between and .

(9)

Now I’ll find out the expression for when is in between and .

###### Step 3

At , the value of is . So one point is .

And at , the value of is . So I’ve got the other point as .

Now I’ll find out the equation of the straight line passing through two points and .

So the equation of the straight line passing through two points and will be

Now I’ll simplify it to get

So this means which gives .

Next, I’ll replace with . Thus the function becomes when is in between and .

In mathematical term, I can say

(10)

At the end, I’ll find out the expression for when is in between and .

###### Step 4

As I can see from the graph, the value of is constant as when is in between and .

(11)

Now I can also notice that the function repeats itself from . So the period of the function is

So I can say that the function has a period of .

(12)

Therefore I can combine equations (8), (9), (10), (11) and (12) to describe the function as

Hence I can conclude that this is the answer to the above-mentioned example.

Dear friends, this is the end of my today’s post on analytical descriptions of functions in Fourier series. 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!!

## Leave a Reply