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). a
Example 1
Coefficients a _{n} is given byAnd coefficients b _{n} is given byA 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 b _{n} = 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. |