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Fourier Series--Square Wave

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FourierSeriesSquareWave

The Fourier series of the odd square wave f(x) of period 2L contains only sine terms. For 0<x<2L, the square wave can be written as

f(x)=2[H(x/L)-H(x/L-1)]-1,
(1)

where H(x) is the Heaviside step function. Since f(x)=-f(2L-x), its periodic extension is odd, so a_0=a_n=0, and

b_n=1/Lint_0^(2L)f(x)sin((npix)/L)dx
(2)

reduces to

b_n=2/Lint_0^Lf(x)sin((npix)/L)dx
(3)
=4/(npi)sin^2(1/2npi)
(4)
=2/(npi)[1-(-1)^n]
(5)
=4/(npi){0 n even; 1 n odd.
(6)

The Fourier series is therefore

f(x)=4/pisum_(n=1,3,5,...)^infty1/nsin((npix)/L).
(7)

See also

Fourier Series, Fourier Series--Sawtooth Wave, Fourier Series--Triangle Wave, Gibbs Phenomenon, Square Wave

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Cite this as:

Weisstein, Eric W. "Fourier Series--Square Wave." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FourierSeriesSquareWave.html

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