Fourier series of functions with period T. Hello friends, today I’ll show how to get the Fourier series of functions with period T. Have a look!!

**Fourier series of functions with period T**

Now the Fourier series of a function with a period is

where are Fourier coefficients and is the angular velocity.

Also, the Fourier coefficients are

where and .

Now I’ll give an example where I’ll get the Fourier series of a function with period .

#### Solved example of the Fourier series functions with period T

Disclaimer: This example DOES NOT belong to me. I have chosen it from some book. I have also given the due reference at the end of the post.

So here is the example.

**Example **

According to Stroud and Booth (2011)*, “Determine the Fourier series representation of the function defined by

”

**Solution**

Now here the given periodic function is

As I can see, it has a period . And I have to get the Fourier series. So I’ll start with the Fourier coefficients and . First of all, I’ll get the value of .

**Step 1**

As I have mentioned above, is

So, in this example it will be

And that means

Now I’ll integrate it using the * standard formulas in integration.* And that gives

Next, I’ll substitute the limits to get

Then I’ll simplify it. And that gives

(1)

**Step 2**

Now I’ll get the value of . As per the formula mentioned above, is

where .

So in this example will be

Therefore will be

Next, I’ll simplify it. So it will be

Now I’ll integrate it. Since , I can say that

Then I’ll simplify it. So that means

And that gives

**Step 3**

Next, I’ll substitute the limits. So I get

Now I’ll simplify it. And that means

Since , I can say that will be

(2)

Now I’ll get the value of .

**Step 4**

As per the formula mentioned above, is

where .

As I have already mentioned in Step 2, will be

Therefore will be

Next, I’ll simplify it. So it will be

Now I’ll integrate it. Since , I can say that

Then I’ll simplify it. So that means

And that gives

**Step 5**

Next, I’ll substitute the limits. So I get

Now I’ll simplify it. And that means

Since , I can say that will be

Then I’ll simplify it. So that means

And that gives

(3)

Now I’ll get the Fourier series .

**Step 6**

As per the formula for the Fourier series, will be

As I can see from equation (2), . Therefore, is also equal to .

Thus I’ll substitute the values of and from equations (1) and (3) to get

Now I’ll replace with and substitute to get the Fourier series as

Next, I’ll simplify it. So that means

As I can see, for the coefficeint of is equal to . So I can say that the Fourier series will be

Now I’ll take out as a common factor. So that gives the Fourier series as

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

Dear friends, this is the end of today’s post on the Fourier series of a function with period 2pi and range [-pi to pi]. 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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