Fourier Series calculated by Mathematica gives wrong result?

cornelbejan15 · February 18, 2023, 2:11pm

Hello,

Why for the function f1 the Fourier series calculated by Mathematica does not give the same graph as the Fourier series calculated with the sum of sin and cos (https://en.wikipedia.org/wiki/Fourier_series)?

In the case of function f2, we can see that the two calculation methods give the same results. I don’t understand why in the case of function f1 the two methods do not give the same results.

I saw that Mathematica gives the Fourier series in the exponential form, but the exponential form and the sine and cos form should give the same results, right?

f1[x_] := \!\(\*
TagBox[GridBox[{
{"\[Piecewise]", GridBox[{
{"0", 
RowBox[{
RowBox[{"-", "5"}], "<", "x", "<", "0"}]},
{"3", 
RowBox[{"0", "<", "x", "<", "5"}]}
},
AllowedDimensions->{2, Automatic},
Editable->True,
GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}},
GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{1.}}},
GridBoxSpacings->{"Columns" -> {
Offset[0.27999999999999997`], {
Offset[0.84]}, 
Offset[0.27999999999999997`]}, "Rows" -> {
Offset[0.2], {
Offset[0.4]}, 
Offset[0.2]}},
Selectable->True]}
},
GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}},
GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{1.}}},
GridBoxSpacings->{"Columns" -> {
Offset[0.27999999999999997`], {
Offset[0.35]}, 
Offset[0.27999999999999997`]}, "Rows" -> {
Offset[0.2], {
Offset[0.4]}, 
Offset[0.2]}}],
"Piecewise",
DeleteWithContents->True,
Editable->False,
SelectWithContents->True,
Selectable->False,
StripWrapperBoxes->True]\)
Plot[f1[x], {x, -5, 5}, Exclusions -> None]
T = 10;
A0 = 2/T*\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(-
\*FractionBox[\(T\), \(2\)]\), 
FractionBox[\(T\), \(2\)]]\(f1[x] \[DifferentialD]x\)\) // Simplify
An = 2/T*\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(-
\*FractionBox[\(T\), \(2\)]\), 
FractionBox[\(T\), \(2\)]]\(f1[x]*Cos[
\*FractionBox[\(2*\[Pi]*n*x\), \(T\)]] \[DifferentialD]x\)\) // 
  Simplify
Bn = 2/T*\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(-
\*FractionBox[\(T\), \(2\)]\), 
FractionBox[\(T\), \(2\)]]\(f1[x]*Sin[
\*FractionBox[\(2*\[Pi]*n*x\), \(T\)]] \[DifferentialD]x\)\) // 
  Simplify
s1[k_, x_] := A0/2 + \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(n = 1\), \(k\)]\((An*Cos[
\*FractionBox[\(2*\[Pi]*n*x\), \(T\)]] + Bn*Sin[
\*FractionBox[\(2*\[Pi]*n*x\), \(T\)]])\)\) // Simplify
Testf1 = FourierSeries[f1[x], x, 40];
Plot[{f1[x], s1[40, x], Testf1}, {x, -5, 5}, Exclusions -> None, 
 PlotLegends -> "Expressions", 
 PlotStyle -> {Thickness[0.008], Thickness[0.005], Thickness[0.003]}]
Plot[{s1[40, x], Testf1}, {x, -5, 5}, Exclusions -> None, 
 PlotLegends -> "Expressions", 
 PlotStyle -> {Thickness[0.008], Thickness[0.005], Thickness[0.003]}]

f2[x_] := \!\(\*
TagBox[GridBox[{
{"\[Piecewise]", GridBox[{
{
RowBox[{"-", "1"}], 
RowBox[{
RowBox[{"-", "\[Pi]"}], "<", "x", "<", "0"}]},
{"1", 
RowBox[{"0", "<", "x", "<", "\[Pi]"}]}
},
AllowedDimensions->{2, Automatic},
Editable->True,
GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}},
GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{1.}}},
GridBoxSpacings->{"Columns" -> {
Offset[0.27999999999999997`], {
Offset[0.84]}, 
Offset[0.27999999999999997`]}, "Rows" -> {
Offset[0.2], {
Offset[0.4]}, 
Offset[0.2]}},
Selectable->True]}
},
GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}},
GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{1.}}},
GridBoxSpacings->{"Columns" -> {
Offset[0.27999999999999997`], {
Offset[0.35]}, 
Offset[0.27999999999999997`]}, "Rows" -> {
Offset[0.2], {
Offset[0.4]}, 
Offset[0.2]}}],
"Piecewise",
DeleteWithContents->True,
Editable->False,
SelectWithContents->True,
Selectable->False,
StripWrapperBoxes->True]\)
Plot[f2[x], {x, -\[Pi], \[Pi]}, Exclusions -> None]
T = 2*\[Pi];
A0 = 2/T*\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(-
\*FractionBox[\(T\), \(2\)]\), 
FractionBox[\(T\), \(2\)]]\(f2[x] \[DifferentialD]x\)\) // Simplify
An = 2/T*\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(-
\*FractionBox[\(T\), \(2\)]\), 
FractionBox[\(T\), \(2\)]]\(f2[x]*Cos[
\*FractionBox[\(2*\[Pi]*n*x\), \(T\)]] \[DifferentialD]x\)\) // 
  Simplify
Bn = 2/T*\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(-
\*FractionBox[\(T\), \(2\)]\), 
FractionBox[\(T\), \(2\)]]\(f2[x]*Sin[
\*FractionBox[\(2*\[Pi]*n*x\), \(T\)]] \[DifferentialD]x\)\) // 
  Simplify
s2[k_, x_] := A0/2 + \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(n = 1\), \(k\)]\((An*Cos[
\*FractionBox[\(2*\[Pi]*n*x\), \(T\)]] + Bn*Sin[
\*FractionBox[\(2*\[Pi]*n*x\), \(T\)]])\)\) // Simplify
Testf2 = FourierSeries[f2[x], x, 40];
Plot[{f2[x], s2[40, x], Testf2}, {x, -\[Pi], \[Pi]}, 
 Exclusions -> None, PlotLegends -> "Expressions", 
 PlotStyle -> {Thickness[0.008], Thickness[0.005], Thickness[0.003]}]
Plot[{s2[40, x], Testf2}, {x, -\[Pi], \[Pi]}, 
 PlotLegends -> "Expressions", 
 PlotStyle -> {Thickness[0.008], Thickness[0.005], Thickness[0.003]}]

Mathematica 13.2 Notebook file attached.

Thank you.

FourierSeries.nb (501 KB)

gianlucagorni · February 18, 2023, 4:16pm

The function FourierSeries assumes by default that the interval is between -Pi and Pi. For other intervals you have to use the option FourierParameters.

cornelbejan15 · February 18, 2023, 4:26pm

Ok. Thank you.

Testf1 = 
  FourierSeries[f1[x], x, 40, 
   FourierParameters -> {-(1/5)*\[Pi], 1/5*\[Pi]}];

cornelbejan15 · February 18, 2023, 4:46pm

Another thing:

When I use subscripts for 0 and n at A0, An, and Bn then it doesn’t calculate… how can I use subscripts at A0, An, and Bn? Is such a thing possible?

In principle, I think I found out why those sums are not calculated when I use subscripts at those variables. For example:

But I’m interested if there is any way to use subscripts for those sums…?

f1[x_] := \!\(\*
TagBox[GridBox[{
{"\[Piecewise]", GridBox[{
{"0", 
RowBox[{
RowBox[{"-", "5"}], "<", "x", "<", "0"}]},
{"3", 
RowBox[{"0", "<", "x", "<", "5"}]}
},
AllowedDimensions->{2, Automatic},
Editable->True,
GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}},
GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{1.}}},
GridBoxSpacings->{"Columns" -> {
Offset[0.27999999999999997`], {
Offset[0.84]}, 
Offset[0.27999999999999997`]}, "Rows" -> {
Offset[0.2], {
Offset[0.4]}, 
Offset[0.2]}},
Selectable->True]}
},
GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}},
GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{1.}}},
GridBoxSpacings->{"Columns" -> {
Offset[0.27999999999999997`], {
Offset[0.35]}, 
Offset[0.27999999999999997`]}, "Rows" -> {
Offset[0.2], {
Offset[0.4]}, 
Offset[0.2]}}],
"Piecewise",
DeleteWithContents->True,
Editable->False,
SelectWithContents->True,
Selectable->False,
StripWrapperBoxes->True]\)
Plot[f1[x], {x, -5, 5}, Exclusions -> None]
T = 10;
Subscript[A, 0] = 2/T*\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(-
\*FractionBox[\(T\), \(2\)]\), 
FractionBox[\(T\), \(2\)]]\(f1[x] \[DifferentialD]x\)\) // Simplify
Subscript[A, n] = 2/T*\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(-
\*FractionBox[\(T\), \(2\)]\), 
FractionBox[\(T\), \(2\)]]\(f1[x]*Cos[
\*FractionBox[\(2*\[Pi]*n*x\), \(T\)]] \[DifferentialD]x\)\) // 
  Simplify
Subscript[B, n] = 2/T*\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(-
\*FractionBox[\(T\), \(2\)]\), 
FractionBox[\(T\), \(2\)]]\(f1[x]*Sin[
\*FractionBox[\(2*\[Pi]*n*x\), \(T\)]] \[DifferentialD]x\)\) // 
  Simplify
s1[k_, x_] := Subscript[A, 0]/2 + \!\(
\*UnderoverscriptBox[\(\[Sum]\), \(n = 1\), \(k\)]\((
\*SubscriptBox[\(A\), \(n\)]*Cos[
\*FractionBox[\(2*\[Pi]*n*x\), \(T\)]] + 
\*SubscriptBox[\(B\), \(n\)]*Sin[
\*FractionBox[\(2*\[Pi]*n*x\), \(T\)]])\)\) // Simplify
Testf1 = 
  FourierSeries[f1[x], x, 40, 
   FourierParameters -> {-(1/5)*\[Pi], 1/5*\[Pi]}];
Plot[{f1[x], s1[40, x]}, {x, -5, 5}, Exclusions -> None, 
 PlotLegends -> "Expressions", 
 PlotStyle -> {Thickness[0.008], Thickness[0.005], Thickness[0.003]}]

Untitled-1.nb (19.6 KB)

gianlucagorni · February 18, 2023, 5:39pm

You have to use patterns: Subscript[A, n_] =, just as in function definitions.

cornelbejan15 · February 18, 2023, 5:56pm

Can you write an example for this line of code?

Subscript[A, n] = 2/T*\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(-
\*FractionBox[\(T\), \(2\)]\), 
FractionBox[\(T\), \(2\)]]\(f1[x]*Cos[
\*FractionBox[\(2*\[Pi]*n*x\), \(T\)]] \[DifferentialD]x\)\)

Do you mean something like this?

gianlucagorni · February 18, 2023, 7:41pm

The pattern, that is, the underscore:

Subscript[A, n_] = 
 2/T Integrate[Exp[x] Cos[2 Pi n x/T], {x, -T/2, T/2}]
Sum[Subscript[A, n], {n, 0, 5}]

Without the underscore, the definition is used only with literal n.

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Fourier Series calculated by Mathematica gives wrong result?

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