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.
According to Croft et al. (2000), “Find the Fourier series representation of the function
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.
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.
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
Now I’ll get the Fourier series .
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!!