Fourier series of a periodic function. Hi there, today I’ll show how to get the Fourier series of a periodic function. Have a look!

Want to read more on how to get the Fourier series of any function? Do check out:

**Fourier series of functions with period 2 pi**

**Fourier series of a function with period 2 pi and range [-pi, pi]**

**How to get the Fourier series of f(x) with x: [0, 2pi)**

**Fourier series of f(x) up to the fourth harmonic**

**Fourier series of functions with arbitrary period**

**How to get the Fourier series of a function with period T**

**How to identify the odd and even functions**

**The Fourier series of an even function**

**Fourier series of a periodic function**

Now here I’ll give an example of a periodic function with a period . Unlike my other examples, here the function has two different values with the same period. Of course, the formula for the Fourier coefficients and the Fourier series will be the same as always.

As we all know, the Fourier series of a function with a period is

where are Fourier coefficients.

Also, the Fourier coefficients are

where .

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

**A solved example of the Fourier series of a periodic function**

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)*, “A function is defined by

Obtain the Fourier series.”

**Solution**

Now here the given periodic function is

As I can see, it has a period of . 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

Thus it will be

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

Then I’ll simplify it. 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

So in this example it will be

Thus it will be

Now I’ll integrate it. And for that, I’ll use the * integration by parts* method. And that says

So in this case, I’ll choose as and as . If , then . Also, for , the value of is

If interested, you can read more on the similar topic of integration like the * reduction formula for cos^n x*,

**reduction formula for e^(-x) cos^n x**.

Thus becomes

Then I’ll simplify it. And that gives

Now I’ll substitute the limits on the first part. Simultaneously, I’ll integrate .

**Step 3**

So that will be

Next, I’ll simplify it. And that means

Now I’ll substitute the limits in . Also, I know that . So the value of will be

If I simplify it, what I get is

Also, I already know that and . Thus the value of will be

(2)

Now I’ll get the value of .

**Step 4**

As per the formula mentioned above, is

So in this example it will be

Thus it will be

Now I’ll integrate it. And for that, I’ll use the * integration by parts* method. And that says

So in this case, I’ll choose as and as . If , then . Also, for , the value of is

Thus becomes

Then I’ll simplify it. And that gives

And that gives

Now I’ll substitute the limits on the first part. Simultaneously, I’ll integrate .

**Step 5**

So that will be

Also, I know that . So the value of will be

Then I’ll simplify it to get

Now I’ll substitute the limits in . So the value of will be

Next, I’ll simplify it. And that gives

Since , I can say that

Therefore the value of will be

(3)

Now I’ll get the Fourier series .

**Step 6**

As per the formula for the Fourier series, will be

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

Then I’ll simplify it. Thus will be

Next, I’ll substitute in to get

Then I’ll simplify it. And that gives

So I can say that

Thus I can say that

is the Fourier series of the given function.

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 periodic function. 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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