Table of Contents

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



More References and Links

Handbook of Mathematical Functions Engineering Mathematics with Examples and Solutions