Tutorials on Fourier series are presented. In the first part an example is used to show how Fourier coefficients are calculated and in a second part you may use an app to further explore Fourier series of the same function. Fourier Series and CoefficientsFourier series may be used to represent periodic functions as a linear combination of sine and cosine functions. If f(t) is a periodic function of period T, then under certain conditions, its Fourier series is given by:
where n = 1 , 2 , 3 , ... and T is the period of function f(t). an and bn are called Fourier coefficients and are given by
Example 1
Solution to Example 1
Coefficients an is given by And coefficients bn is given by A computation of the above coefficients gives \( a_0 = 0 \), \( a_n = 0 \) and \( b_n = \dfrac{2}{n\pi} (1 - \cos (n \pi)) \) Note that \( \cos (n \pi) \) may be written as \( cos (n \pi) = (-1)^n \) and that bn = 0 whenever n is even. The given function f(t) has the following Fourier series Interactive Tutorial on Fourier Series
For numerical calculations purposes we cannot include an infinite number of terms in the series above, we therefore define function \( f_N(t) \) with a limited number of terms \( N \) as follows
\[ f_N(t) = \sum_{n=1}^{N} \dfrac{2}{n\pi} (1-(-1)^n) \sin(\dfrac{2 n \pi t}{T}) \]
The app below may be used to explore the Fourier series of f(t) solved in example 1 above including a limited number of terms \( N \) in the series and see how the graph of function \( f_N(t) \) defined above becomes close to the graph of function f(t) as \( N \) increases.
Hover the mousse cursor on the graph or plotted point to read the coordinates. |