Fourier series calculator

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

a02n1ancosnπxkbnsinnπxk

where

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:

an11xcosnπxdx0

bn11xsinnπxdx21nnπ

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

n121nnπsinnπx

We can see two plots on the figure below f(x)=x (yellow 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]:

an1kqpfxcosnπxkdx

bn1kqpfxsinnπxkdx

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
Find fourier series of the function fxx2on interval [π;π]by common formula.Order of expansion 5.
Function to find Fourier series expansion


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