Fourier series of even functions. Hello friends, today I’ll show how to get the Fourier series of even functions. Have a look!

If interested, you can check out on * how to determine the odd and even functions*.

**Fourier series of even functions**

Now for the Fourier series of even functions, there’s a very handy rule. And it says suppose an even function is symmetrical about the -axis. Then the Fourier series of that function will have only and cosine terms.

And the term ‘symmetrical about the -axis’ means ”the range of can be in between to or to – something like that.

So the Fourier series of an even function with a period is

where are Fourier coefficients and is the angular velocity.

And the Fourier coefficients are

where and .

Now I’ll show you an example.

**A solved example of the Fourier series of even functions**

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 defined by

Determine its Fourier series.”

**Solution**

Now here the given function is

First of all, I’ll check if it’s an odd or even function. So I’ll replace with . And that gives

As I can see, . So the function is an even function. Therefore the Fourier series of will have only and terms. Now I’ll get the value of .

**Step 1**

Since , I can say that the period . Therefore will be

Thus will be

Since is an even function and is defined in between and , I can rewrite as

Now I’ll integrate it using the * standard formulas in integration*. So it will be

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

Next, I’ll simplify it. So it will be

Now I’ll integrate it. As I can see, the integration of the first expression is straight forward. But for the second one, 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

**Step 3**

Thus becomes

Then I’ll simplify it. And that gives

Now I’ll substitute the limits in the first two terms. Also, I’ll integrate the last term. Again I’ll use the integration by parts method for that. So I’ll choose as and as . If , then . Also, for , the value of is

Thus becomes

Next, I’ll simplify it.

**Step 4**

So I get

As I know already . So becomes

Then I’ll substitute the limits in . Simultaneously, I’ll integrate . Thus it becomes

So I can say that

Also, I know that . Now I’ll substitute the limits in . And that gives

Next, I’ll simplify it to get

Since , I can say that

(2)

Next, I’ll get the Fourier series.

**Step 5**

As per the formula for the Fourier series of even functions, will be

So I’ll substitute the values of and from equations (1) and (2) to get the value of . Also, I have already mentioned the value of as in Step 1.

Thus will be

Then I’ll simplify it. So will be

Next, I’ll substitute to get

Then I’ll simplify it. And that gives

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 even functions. 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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