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Fourier Series--Sawtooth Wave

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FourierSeriesSawtoothWave

Consider a string of length 2L plucked at the right end and fixed at the left. The functional form of this configuration is

f(x)=x/(2L).
(1)

The components of the Fourier series are therefore given by

a_0=1/Lint_0^(2L)x/(2L)dx
(2)
=1
(3)
a_n=1/Lint_0^(2L)x/(2L)cos((npix)/L)dx
(4)
=([2npicos(npi)-sin(npi)]sin(npi))/(n^2pi^2)
(5)
=0
(6)
b_n=1/Lint_0^(2L)x/(2L)sin((npix)/L)dx
(7)
=(-2npicos(2npi)+sin(2npi))/(2n^2pi^2)
(8)
=-1/(npi).
(9)

The Fourier series is therefore given by

f(x)=1/2-1/pisum_(n=1)^(infty)1/nsin((npix)/L)
(10)
=1/2+i/(2pi)ln(-e^(-ipix/L))
(11)
=1/2-1/(2pi)arg(-e^(-ipix/L)).
(12)

See also

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

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References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 762-763, 1985.

Cite this as:

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

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