On-Line Fourier Series Calculator is an interactive app to calculate Fourier Series coefficients (Up to 10000 elements) for user-defined piecewise functions up to 5 pieces, for example.
\( f(x) = \left\{\begin{matrix} 0 & x \in [-1,0)\\ x+1 & x \in [0,1] \end{matrix}\right. \)
Produces the result
Note that function must be in the integrable functions space or L
1 on selected Interval as we shown at theory sections.
FourierSeries Calculator calculates Fourier Coefficients, analytic and numeric integrals and it is usefull to plot 1-variable functions and its Fourier series on a generic user-defined interval.
Click here to access to Fourier Series Calculator
Calculations accuracy depends largely of size-interval introduzed and number of selected coefficients to calculate.
Use it is as follows.
1) Write the lower end of the range in the text box labeled Limit inf.
2) Enter the upper range in the text box labeled Limit Sup.
3) Write the function in the text box with the label function
In the piecewise function case, operate as follows
1) Write the lower end of the range in the text box labeled Limit inf.
2) Enter the upper range in the text box labeled Limit Sup.
3) Write the first function in the text box with the label function
.
4) Enter the upper sub-range in the text box labeled Subinterval 1.
5) Write the function as defined in the first sub-interval in the text box labeled subinterval 1
.
For example suppose we have the piecewise function
\( f(x) = \left\{\begin{matrix} 0 & x \in [-1,0)\\ x+1 & x \in [0,1] \end{matrix}\right. \)
Then the fields are filled as
After the A
n, B
n calculations, is possible to plot the function and its Fourier Series by clicking "Show Graph". At this case
By default, the problem begins with the continuous function and the interval following
\( f(x) = \left\{\begin{matrix} x^2+x+1 & x \in [-1,0)\\ 0 & x \in [0,1] \end{matrix}\right. \)
How it works?
To calculate Fourier coefficients we apply integration methods seen in the
numerical methods section, with which we can approximate the integrals given by the Fourier Series formulae:
$$ f(x)=\frac{A_{0}}{2} + \sum_{n=1}^{\infty}(A_n cos nx + B_nsin nx ) $$
Where
\( A_i, B_i \in \mathbb{R} \)
and
\( A_0= \frac{1}{\pi} \int_{-\pi}^{\pi}f(x) dx\)
\( A_n= \frac{1}{\pi} \int_{-\pi}^{\pi}f(x) cos nx dx\)
\( B_n= \frac{1}{\pi} \int_{-\pi}^{\pi}f(x) sin nx dx\)
In Fourier coefficients case, there are several methods to make the calculations programmed here by the owners of
Mathstools .
To calculate the derivative of the function: uses severeal numerical methods to derivate.
To calculate the primitive function: numerical integration methods seen in the
numerical methods section are applied.
Note that in numerical analysis, errors are obtained due to the particular methods and the limitation of arithmetic computer as well.
In the Fourier coefficients calculations case, it depends on the function and size of the chosen integration interval. In default problem
the error in calculating the Fourier coefficients is O (1e-8). For the numerical integration is O (1e-11) and in the derivative it is O (1e-14).
Click here to access to Fourier Series Calculator
