-order Fourier sine series expansion of 
FourierSinSeries
FourierSinSeries[expr,t,n]
gives the n
-order Fourier sine series expansion of expr in t.
FourierSinSeries[expr,{t1,t2,…},{n1,n2,…}]
gives the multidimensional Fourier sine series of expr.
Details and Options
- The
-order Fourier sine series of 
is by default defined to be 
with 
. - The
-dimensional Fourier sine series of 
is given by 
with 
. - The following options can be given:
-
Assumptions $Assumptions assumptions on parameters FourierParameters {1,1} parameters to define Fourier sine series GenerateConditions False whether to generate results that involve conditions on parameters - Common settings for FourierParameters include:
-
{1,1} 
{1,2Pi} 
{a,b} 
- The Fourier sine series of
is equivalent to the Fourier series of 
.
Examples
open all close allBasic Examples (2)
Find the 5
-order Fourier sine series approximation to t:
FourierSinSeries[t, t, 5]Plot[%, {t, -3π, 3π}]Find the 3
-order bivariate Fourier sine series approximation to 
:
FourierSinSeries[x y, {x, y}, {3, 3}]Plot3D[%, {x, -Pi, Pi}, {y, -Pi, Pi}]Scope (3)
Find the 3
-order Fourier sine series approximation to a quadratic polynomial:
FourierSinSeries[t ^ 2 + 3t + 1, t, 3]Fourier sine series for a piecewise function:
FourierSinSeries[UnitStep[t(π / 2 - t)], t, 14]Plot[%, {t, 0, Pi}, PlotRange -> All]The Fourier sine series for a basis function has only one term:
FourierSinSeries[Sin[3 t], t, 5]Options (1)
FourierParameters (1)
Use a nondefault setting for FourierParameters:
FourierSinSeries[UnitBox[t], t, 10, FourierParameters -> {1, 2π}]Plot[%, {t, 0, 1 / 2}, PlotRange -> All]Properties & Relations (1)
fs = FourierSinSeries[t ^ 2, t, 3]The Fourier series of the odd extension of 
:
f = FourierSeries[Piecewise[{{t ^ 2, t ≥ 0}, {-t ^ 2, t < 0}}], t, 3]In general these will always coincide:
Simplify[f - fs]The Fourier sine series of 
approximates 
:
Plot[{fs, Piecewise[{{t ^ 2, t ≥ 0}, {-t ^ 2, t < 0}}]}, {t, -π, π}]Text
Wolfram Research (2008), FourierSinSeries, Wolfram Language function, https://reference.wolfram.com/language/ref/FourierSinSeries.html.
CMS
Wolfram Language. 2008. "FourierSinSeries." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FourierSinSeries.html.
APA
Wolfram Language. (2008). FourierSinSeries. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FourierSinSeries.html