# 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
$$\displaystyle f(t) = \dfrac{1}{2 \pi} \int_{-\infty}^{+\infty} F(\omega) \exp \left( j \omega t\right) d\omega$$