Using the Fourier Series, we simply decompose a particular real-valued function into terms containing the sine & cosine functions.

This function, lets say f(x) is taken to be periodic with a period 2pi. We could further extent the definition of the Fourier Series for functions with arbitrary periods.

The Fourier Series of a function f(x) could we written down as...

In the above decomposition, are called Fourier Coefficients or Euler's Coefficients.

The former because they appear in the Fourier Series & the latter because we obtain their values using the Euler's Formula.

Now the Euler's Formula gives the the values of the Euler's coefficients & for a function with period 2pi, we have...

Now, once you know the values of the coefficients(which would come out to be functions of n) you could find out the complete sum & hence, the complete Fourier series of the function f(x).

The value of 'c' here would depend on the interval specified with the function f(x).

For example, Lets say we have a function,

on the interval...

We have here a simple saw tooth function which we've made periodic with a period 2pi.

To start with, we have the interval correpsonding to c to c+2pi which gives c=-pi.

Hence, the values of the coefficients would now be given by...

Hence, on integration we obtain...

This would generate the following Fourier series for our function f(x)...