# Fourier Series: Example 3 - Parabolic Function

Complex Exponential Form Solution

**Figure 1: Periodic Parabolic function**

## Trigonometric Form Solution

The coefficients for the Trigonometric Form of the Fourier Series are:

1)

2)

3)

Using the above coefficients, we can calculate the Fourier Series of f(t):

**Figure 2: Harmonics Addition for Parabolic function**

## Harmonic Form Solution

The coefficients are:

So the Fourier Series for the function:

This is the same like the Trigonometric Form, as expected, since there are no sine factors.

## Complex Exponential Form Solution

The coefficients are:

Now, because:

It is:

Also for n=0 it is:

**Figure 3: Amplitude and Phase for Complex Exponential Coefficients**

The Fourier Series for the function f(t):

Now because:

It is: