Discovering the Fourier Series of a General Expression: A Comprehensive Guide

[crazy_nuttie]

How can I represent a general "expression" as a Fourier series?

For example, I want to find the Fourier series of sum:
$$\frac{1}{1^2}$$ + $$\frac{1}{3^2}$$ + $$\frac{1}{5^2}$$ ... (infinite).

using the value of the Fourier series at x = 0 (because this will give the value of the infinite sum).

I'm not sure how I can put this into a function, and find the Fourier series (for example, what would be the "p" value of the function?)

Thanks

 

[AlephZero]

If you take the Fourier series
F(x) = cos(x) + cos(3x)/3^2 + cos(5x)/5^2 + ...
then F(0) = 1 + 1/3^2 + 1/5^2 + ...

So you want to find what the function F(x) is.

This is a similar problem to integrating a function. There isn't any "plug and chug" way to do integration. You have use the fact that integration is the same as anti-differentiation, and transform the integral into a form where you already "know the answer".

One way to do this for Fourier series is draw a graph of the first few terms of the series and see what it looks like. Then guess an expression that might represent the function, find its Fourier series, and see if you guessed right.

Another "trick" is that you can integrate and differentiate Fourier series term by term. So, the differential of your function is

F'(x) = -(sin(x) + sin(3x)/3 + sin(5x)/5 + ...)

You might recognize that as the Fourier series of something you have seen before. If not, draw the graph of the sum of the first 5 or 10 terms, and it might help you guess what F(x) is.