Fourier series calculator

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Expansion of some function f(x) in trigonometric Fourier series on interval [-k, k] has the form:

Fourier series expansion formula

where

an coefficients of Fourier series expansion formula для (n = 0, 1, 2, 3,...)

bn coefficients of Fourier series expansion formula для (n = 1, 2, 3,...)

As an example, find Fourier series expansion of the function f(x)=x on interval [-1, 1]. In this case the coefficients an и bn are determined by the formulas:

formula to calculate an coefficients of Fourier series on interval [-1;1]

formula to calculate bn coefficients of Fourier series on interval [-1;1]

Therefore, the expansion of function f(x)=x in Fourier series on interval [-1, 1] has the form:

Fouries series of the function f(x)=x on interval [-1;1]

We can see two plots on the figure below f(x)=x (yellow color) and Fouries series of the function f(x)=x on interval [-1;1] , (blue color) for which we use order of expansion equal to 25.

plot of the у=x function and its Fourier series expansion of 25 order

It should be noted, that in example above, the coefficients an are zero not by chance. The fact is that function f(x)=x is odd on interval [-1, 1]. In contrast, the function функция cos(n*pi*x) - is even. The product of an even function by the odd one is the odd function, so according to the properties , integral of the odd function on symmetric interval is zero.

In case of the even function, for example x2 , coefficients bn were zero, because the integrand integrand - is odd function.

Based on the above reasoning, we can draw the following conclusions:

It should be noted, that by using the formulas given above and corresponding variable substitution, it is possible to obtain the formulas for Fourier series expansion coefficients of some function at an arbitrary interval. [p, q]:

formula for Fourier series expansion coefficients a(n) of some function at an arbitrary interval

formula for Fourier series expansion coefficients b(n) of some function at an arbitrary interval

here k=(q-p)/2 .

Our online calculator, build on Wolfram Alpha system finds Fourier series expansion of some function on interval [-π π]. In principle, this does not impose significant restrictions because using the corresponding variable substitution we can obtain an expansion at an arbitrary interval [p, q].

Fourier series calculator
Function to find Fourier series expansion
Function to find Fourier series expansion



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