Fourier Series—Semicircle

Author

Eric W. Weisstein
August 17, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/FourierSeriesSemicircle.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/FourierSeriesSemicircle.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Figure

V6

<<Utilities`Typesetting`

FourierPartialSums[(Sqrt[(2L-x) x]),{x,0,2L},3,Assumptions->L>0]//FullSimplify//TraditionalForm

FourierPlot[(Sqrt[(2-x) x]),{x,0,2},5,AxesLabel->TraditionalForm/@{InlineFraction[x,L],(Sqrt[(2L-x) x])}]

Integral

FullSimplify[(Sqrt[(a^2)-(((x-L))^2)])/.a->L]

With[{L=1},Plot[(Sqrt[(2 L-x) x]),{x,0,2L},PlotRange->All,AspectRatio->Automatic,PlotStyle->Red]
]

Simplify[FourierA[0,(Sqrt[(2 L-x) x]),{x,0,2L}],L>0]

Simplify[FourierA[n,(Sqrt[(2 L-x) x]),{x,0,2L}],L>0&&n≥1]

Simplify[FourierB[n,(Sqrt[(2 L-x) x]),{x,0,2L}],n≥1&&L>0]

Plot[Pi/4+Sum[(((((-1))^n) BesselJ[1,n π])/n)Cos[n x Pi],{n,30}],{x,0,2},PlotRange->{{0,2},{0,1.1}},PlotStyle->Red,AspectRatio->Automatic]

Plot[Pi/4+Sum[(((((-1))^n) BesselJ[1,n π])/n)Cos[n x Pi],{n,30}],{x,0,4},PlotRange->{{0,4},{0,1.1}},PlotStyle->Red,AspectRatio->Automatic]

V7

FourierCosCoefficient[(Sqrt[(2-x) x]),x,n]

Table[FourierCosCoefficient[(Sqrt[(2-x) x]),x,n],{n,0,1}]

FullSimplify[((π+I (Sqrt[-2+π]) (π^(3/2))-I ((Sqrt[(-2+π) π])+Log[-1+π+(Sqrt[(-2+π) π])]))/π)]

N[%]

FourierSinCoefficient[(Sqrt[(2-x) x]),x,n]

Table[FourierSinCoefficient[(Sqrt[(2-x) x]),x,n],{n,1}]

Sum

Pi/4+Sum[(((((-1))^n) BesselJ[1,n π])/n)Cos[n x Pi],{n,∞}]//InputForm

f=Pi/4+Sum[(((((-1))^n) BesselJ[1,n π])/n)Cos[n x Pi],{n,100}];

Plot[f,{x,0,4},PlotRange->{0,1},AspectRatio->Automatic]

Plot[(Sqrt[(2-x) x]),{x,0,2},AspectRatio->Automatic]

Plot[f-(Sqrt[(2-x) x]),{x,0,2},WorkingPrecision->20]