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

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

Also, you can look out for **how to get the Fourier series for even functions**

**Fourier series of odd functions**

Now for the Fourier series of odd functions, there’s a very handy rule. And it says suppose an odd function . Then the Fourier series of that function will have only sine terms.

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

where is the Fourier coefficient and is the angular velocity.

And the Fourier coefficient is

where and .

Now I’ll show you an example.

**A solved example of the Fourier series of odd 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 Croft et al. (2000), “Find the Fourier series representation of the function

”

**Solution**

Now here the given periodic function is

”

As I can see, is an odd function. So I can get the Fourier series of an odd function.

**Step 1**

Now the period is . So the angular velocity is

Since is an odd function, the Fourier coefficients and will be . Thus the other Fourier coefficient will be

Since and , will be

Thus will be

Next, I’ll simplify it. And that gives

Then I’ll integrate it using the **standard formulas in integration.**

**Step 2**

And that gives

Now I’ll simplify it to get

Then I’ll substitute the limits. And that gives

Next, I’ll simplify it. So I get

And this is because .

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

Then I’ll simplify it. And that means

So this gives

(1)

Now I’ll get the Fourier series .

**Step 3**

As per the formula for the Fourier series, will be

So I’ll substitute the values of from equation (1) and to get

Then I’ll simplify it. Thus will be

Next, I’ll substitute in to get

Now I’ll simplify it. And that gives

Therefore I can say that

Next, I can take out as a common term. So it will be

Thus I can say that

is the Fourier series of the given function.

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

Dear friends, this is the end of today’s post on the Fourier series of odd 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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