# Formulas For Fourier Series and Transform

## Real Fourier Series

If $f(t)$ is a periodic function with period $T$, then
$\displaystyle f(t) = \dfrac{1}{2} a_0 + \sum_{m=1}^{\infty} a_m \cos \left(\dfrac{2 \pi m}{T} t\right) + \sum_{m=1}^{\infty} b_m \sin \left(\dfrac{2 \pi m}{T} t\right)$

$\displaystyle a_m = \dfrac{2}{T} \int_0^T f(t) \cos \left(\dfrac{2 \pi m}{T} t\right) dt$

$\displaystyle b_m = \dfrac{2}{T} \int_0^T f(t) \sin \left(\dfrac{2 \pi m}{T} t\right) dt$

## Complex Fourier Series

$j = \sqrt{-1}$ is the imaginary unit
If $f(t)$ is a periodic function with period $T$, then
$\displaystyle f(t) = \sum_{m = -\infty}^{\infty} c_m \exp \left( j \dfrac{2 \pi m}{T} t \right)$

$\displaystyle c_m = \dfrac{1}{T} \int_0^T f(t) \exp \left( - j \dfrac{2 \pi m}{T} t\right) dt$

## Relationship Between Real and Complex Coefficients

$c_m = \dfrac{1}{2} (a_m - j b_m) , m \gt 0$
$c_0 = \dfrac{1}{2} a_0$
$c_m = \dfrac{1}{2} (a_{-m} + j b_{-m}) , m \lt 0$

## Fourier Transform Pair

If $f(t)$ is defined in the range $-\infty \lt t \lt +\infty$, then the fourier transform $F(\omega)$ is defined by
$\displaystyle F(\omega) = \int_{-\infty}^{+\infty} f(t) \exp \left( - j \omega t\right) dt$
and
$\displaystyle f(t) = \dfrac{1}{2 \pi} \int_{-\infty}^{+\infty} F(\omega) \exp \left( j \omega t\right) d\omega$