Fourier series of f(x) with x:[0 to 2 pi). Hello friends, today I’ll talk about the Fourier series of a function with period 2pi and range [0 to 2 pi). Have a look!

If you want to know more on the Fourier series of other functions, do check out:

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

**How to get the Fourier series of a function of period 2 pi and range [-pi, pi]**

**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 f(x) with x:[0 to 2 pi)

Now 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 function with period and range .

#### Solved example of the Fourier series of a function with period 2pi and range [0 to 2 pi]

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

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

So in this example 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

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

**Step 3**

Now, for the integration of , I’ll do it in the same way as in Step 2. So I’ll choose as and as . If , then . Also, for , the value of is

Thus becomes

As I know, . Thus will be

Now I’ll simplify the other part. And that gives

Next, I’ll substitute the limits. Also, I’ll integrate . Thus it will be

Then I’ll simplify it. Since , I can say that

Next, I’ll substitue the limits to get

Also, . Thus will be

Then I’ll simplify it. And that gives

(2)

Now I’ll get the value of .

**Step 4**

As per the formula mentioned above, is

So in this example 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

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

**Step 5**

Now, for the integration of , I’ll do it in the same way as in Step 4. So I’ll choose as and as . If , then . Also, for , the value of is

Thus becomes

Since , I can say that

Next, I’ll simplify it. And that gives

Then I’ll substitute the limits on the first part. Also, I’ll integrate .

**Step 6**

Thus it will be

Since , I can say that

Then I’ll simplify it and substitute the limits on the second part. So that gives

Since , I can say that

Thus the value of will be

(3)

Now I’ll get the Fourier series .

**Step 7**

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

And this 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 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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