Fourier series calculator

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



an1kkkfxcosnπxkdx для (n = 0, 1, 2, 3,...)

bn1kkkfxsinnπxkdx 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:



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


We can see two plots on the figure below f(x)=x (red color) and yx25n121nnπsinnπx , (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 cosnπx - 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 x2sinnπx - 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]:



here kqp2 .

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
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Find fourier series of the function fxx2on interval [0,3]only by cosines.Order of expansion is 10.

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See also:

Series convergence calculator
Inverse Laplace transform online
Laplace transform online

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