Fourier series of f(x) up to the fourth harmonic. Hello friends, today I’ll talk about the Fourier series of a function up to the fourth harmonic. Have a look!

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

**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 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) up to the fourth harmonic**

Now the Fourier series of a function with a period is

where are Fourier coefficients.

Also, the Fourier coefficients are

where .

Now the first harmonic or the fundamental harmonic is and the th harmonic is . So fourth harmonic means .

Now I’ll give an example where I’ll get the Fourier series of a function up to the fourth harmonic.

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

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 up to the fourth harmonic.”

**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

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. Now I can cancel from both the numerator and the denominator in second and fourth terms. And that gives

Then I’ll simplify it a bit more to get

(1)

**Step 2**

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

So in this example it will be

First of all, I’ll simplify it to get

Now I’ll integrate it.

As I can see, it’s straight forward to integrate but not . And for that, I’ll use the * integration by parts* method. And that says

So in this example, 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**.

**Step 3**

Thus becomes

Then I’ll simplify it. And that gives

Now I’ll integrate the last term to get

So that means

Next, I’ll substitute the limits.

**Step 4**

And that gives

Also, I know that and . And the values of is . So will be

Now I’ll simplify it to get

(2)

Then I’ll get the value of .

**Step 5**

As per the formula mentioned above, is

So in this example it will be

First of all, I’ll simplify it to get

Now I’ll integrate it. As I can see, it’s straight forward to integrate but not . 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

**Step 6**

Thus becomes

Then I’ll simplify it. And that gives

Now I’ll integrate the last term to get

So that means

Next, I’ll substitute the limits.

**Step 7**

And that gives

Also, I know that and . And the values of is and is . So will be

(3)

Now I’ll get the Fourier series .

**Step 8**

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

Also, I can rewrite it as

Now I’ll substitute to get the value of as

Since I have to get the Fourier series up to the fourth harmonic, so I can get up to . Thus I can say that

is the Fourier series of the given function up to the fourth harmonic.

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 up to the fourth harmonic. 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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