# Fourier Series

#### Fourier Series for Real-valued Functions

_{a}

_{0}

_{n}

_{n}

Each nω_{0} is a **harmonic** of the fundamental frequency.
$$f\left(t\right)={v}_{a}+\underset{n=1}{\overset{\infty}{\Sigma}}\left({a}_{n}cos\left(n{\omega}_{0}t\right)+{b}_{n}sin\left(n{\omega}_{0}t\right)\right)$$

Where: $${\omega}_{0}=\frac{2\pi}{T}=\frac{\tau}{T}$$ $${v}_{a}=\frac{1}{T}\underset{{t}_{0}}{\overset{{t}_{0}+T}{\int}}f\left(t\right)dt$$ $${a}_{n}=\frac{2}{T}\underset{{t}_{0}}{\overset{{t}_{0}+T}{\int}}f\left(t\right)cos\left(n{\omega}_{0}t\right)dt$$ $${b}_{n}=\frac{2}{T}\underset{{t}_{0}}{\overset{{t}_{0}+T}{\int}}f\left(t\right)sin\left(n{\omega}_{0}t\right)dt$$

#### Fourier Series for Complex Valued Functions

_{n}

Where: $${C}_{n}=\frac{1}{T}\underset{{t}_{0}}{\overset{{t}_{0}+T}{\int}}f\left(t\right){e}^{-in{\omega}_{0}t}dt$$

A **Fourier Series** is a representation of a periodic function using an infinite summation of sines and cosines. If a finite summation is used the resulting expression will approximate the function. The approximation can be made to any necessary precision by carrying out a larger summation.

The necessary conditions for a function to have a Fourier transform are not known, however **Dirichlet's Conditions** are a set of sufficient conditions for the existence of the Fourier series:

- f(t) has only one value for any t
- f(t) has a finite number of discontinuities in its period
- f(t) has a finite number of minima and maxima in its period
- $\underset{{t}_{0}}{\overset{{t}_{0}+T}{\int}}\left|f\left(t\right)\right|dt$ is finite

If the function f(t) is even then the odd portion of the Fourier series is 0 such that b=0. Likewise if f(t) is odd then a=0 and a_{v}=0. Because of superposition, if the function can be made odd by shifting it along the dependent axis then a=0 still but a_{v} must still be calculated. There are also simplifications if f(t) has half- or quarter-wave symmetry.