**Branch :**Computer Science and Engineering

**Subject :**Math -2

## Properties of the Fourier Series

**Properties of the Fourier Series:**

Basic properties of Fourier Series which are given as follows:

**Linearity:**

F is a linear transformation and so superposition holds. In other words, assume that a and b are simple real numbers, that x and y are T-periodic functions, and that

**Symmetry:**

**1. Even waveforms:**

A waveform x is even if x (t) = x (-t)

If x is also a T-periodic function and F (x) = X, then

for all k.

**2. Odd Waveforms:
**

A waveform x is odd if x (t) = - x (-t)

If x is also a T-periodic function and F (x) = X, then

**3. Real Waveforms:
**

If x is a real value T-periodic function and F (x) = X then

X_{k} = X_{-k }* for all k. In other word, X_{k} equals the complex of X_{-k}. As a result part of X is even and the imaginary part is odd.

**Differentiation:
**

Let x be a T-periodic functions and

for all t. If F (x) = X and F (y) = Y, then

for all k.

**Integration:
**

Let x be a T-periodic functions and

for all t. If F (x) = X and F (y) = Y, then

For all k. In this case X_{0} must be zero, otherwise the integral would not be a periodic function and its Fourier series would not exist.