Plot defined function several times

Question

I want to plot the partial Fourier series of a function. I have this:

f[x_] := x; 
T = 2;
a0 = 2/T*Integrate[f[x], {x, -1, 1}];
a[k_] := 2/T*Integrate[f[x]*Cos[2*Pi*k/T*x], {x, -1, 1}];
b[k_] := 2/T*Integrate[f[x]*Sin[2*Pi*k/T*x], {x, -1, 1}];
g[x_, n_] := 
  a0/2 + Sum[a[k]*Cos[2*Pi*k/T*x], {k, 1, n}] + 
   Sum[b[k]*Sin[2*Pi*k/T*x], {k, 1, n}];
g[x, 3]
Plot[%, {x, -1, 1}, PlotRange -> {-1, 1}, AspectRatio -> 1]

I want to plot the graphs for several of them, with 'n' varying. If posible with 'Animate'.

Answer 1

f[x_] := x;
T = 2;
a0 = 2/T*Integrate[f[x], {x, -1, 1}];
a[k_] := 2/T*Integrate[f[x]*Cos[2*Pi*k/T*x], {x, -1, 1}];
b[k_] := 2/T*Integrate[f[x]*Sin[2*Pi*k/T*x], {x, -1, 1}];
g[x_, n_] := 
  a0/2 + Sum[a[k]*Cos[2*Pi*k/T*x], {k, 1, n}] + 
   Sum[b[k]*Sin[2*Pi*k/T*x], {k, 1, n}];

To look at all plots at once

ParametricPlot3D[
 Evaluate[{x, #, g[x, #]} & /@ Range[10]],
 {x, -1, 1},
 BoxRatios -> {1, 1, 1/2},
 AxesLabel -> (Style[#, 14, Bold] & /@
    {"x", "n", "g "})]

To selectively look at each individual plot

Manipulate[
 Plot[Evaluate@g[x, n], {x, -1, 1},
  AxesLabel -> (Style[#, 14, Bold] & /@
     {"x", "g"}),
  PlotStyle -> ColorData[97][n]],
 {{n, 3}, Range[10], ControlType -> SetterBar}]

Answer 2

f[x_] := x; 
T = 2;
a0 = 2/T*Integrate[f[x], {x, -1, 1}];
a[k_] := 2/T*Integrate[f[x]*Cos[2*Pi*k/T*x], {x, -1, 1}];
b[k_] := 2/T*Integrate[f[x]*Sin[2*Pi*k/T*x], {x, -1, 1}];
g[x_, n_] := 
  a0/2 + Sum[a[k]*Cos[2*Pi*k/T*x], {k, 1, n}] + 
   Sum[b[k]*Sin[2*Pi*k/T*x], {k, 1, n}];
Manipulate[
 Plot[Evaluate[g[x, n]], {x, -1, 1}, PlotRange -> {-1.1, 1.1}, 
  AspectRatio -> 1], {n, 1, 10, 1}]

Answer 3

f[x_] := x;
T = 2;
a0 = 2/T*Integrate[f[x], {x, -1, 1}];
a[k_] := 2/T*Integrate[f[x]*Cos[2*Pi*k/T*x], {x, -1, 1}];
b[k_] := 2/T*Integrate[f[x]*Sin[2*Pi*k/T*x], {x, -1, 1}];
g[x_, n_] := 
  a0/2 + Sum[a[k]*Cos[2*Pi*k/T*x], {k, 1, n}] + 
   Sum[b[k]*Sin[2*Pi*k/T*x], {k, 1, n}];
g[x, 3]
Plot[Tooltip@Table[g[x, n], {n, 5}] // Evaluate, {x, -1, 1}, 
 PlotRange -> {-1, 1}, AspectRatio -> 1, PlotLegends -> Automatic]
Manipulate[
 Plot[Tooltip@Table[g[x, n], {n, i}] // Evaluate, {x, -1, 1}, 
  PlotRange -> {-1, 1}, AspectRatio -> 1, 
  PlotLegends -> Automatic], {i, Range[5]}]

Answer 4

f[x_] := x;
T = 2;
(*
a0 = 2coe[f, Cos, T, 0]
a[k_] := coe[f, Cos, T, k]
b[k_] := coe[f, Sin, T, k]
*)
coe[func_, type: Cos | Sin, per_, order_Integer] := coe[func, type, per, order] = Module[
    {min = -per/2, max = per/2, factor = If[order == 0, 1/2, 1], x},
    Return[
        (2factor)/per Integrate[func[x] type[(2\[Pi] order)/per x], {x, min, max}]
]];
(*
g[x_, n_] := partsum[f, T, n][x]
*)
partsum[func_, per_, order_Integer] := partsum[func, per, order] = Module[
    {},
    x\[Function]
        Sum[
            coe[func, Sin, per, i] Sin[(2\[Pi] i)/per x] + 
            coe[func, Cos, per, i] Cos[(2\[Pi] i)/per x], 
        {i, 0, order}]
];
(* Interpolation *)
partsumtrans[func_, per_, order1_Integer, order2_Integer, tall_] 
    /;order2>order1 := 
    partsumtrans[func, per, order1, order2, tall] = 
    Module[{nth = order2 - order1},
        t\[Function](
            x\[Function](
                (1-SawtoothWave[nth t/tall])
                partsum[func, per, 
                     order1+Floor[t/tall nth]][x]+
                SawtoothWave[nth t/tall]
                partsum[func, per, 
                     order1+Floor[t/tall nth]+1][x]))
]

Now we can use them to construct a demonstration:

Manipulate[Plot[partsumtrans[f, T,  0, 3, 10][t][x], {x, -1, 1}, 
    PlotRange->1, 
    PlotLegends->If[IntegerQ@(3/10t), "n="<>ToString[0+3/10t],
        "n="<>ToString[Floor[0+3/10t]]<>
        " to "<>"n="<>ToString[Ceiling[0+3/10t]]], 
    PlotLabel->TraditionalForm[
        FullSimplify[partsumtrans[f, T, 0, 3, 10][t][x]]]
    ], {t, 0, 10}, 
    AppearanceElements->All,
    ContentSize->{500, 460},
    Alignment->{0, 0}]

@Bob Hanlon Style:

Show[ParametricPlot3D[
    Evaluate[{x, 2#/10, partsum[f, T, #][x]}&/@Range[0, 10]],
    {x, -1, 1},
    PlotStyle->Thick
],
ParametricPlot3D[
    {x, y, partsumtrans[f, T, 0, 10, 2][y][x]},
    {x, -1, 1}, {y, 0, 2},
    Exclusions->None,
    Mesh->None, PlotStyle->Directive[Opacity[.5]],
    ColorFunction->"Rainbow",
    PlotPoints->30
]]

Or like @cvgmt's:

DynamicModule[{t},
    Column[{Dynamic@Plot[
        Evaluate[{partsumtrans[f, T, 0, 5, 1][t][x]}~Join~
        (partsum[f, T, #][x]&/@Range[0,5])],
        {x, -1, 1},
        PlotRange->{{-1, 1}, {-1.2, 1.2}},
        AspectRatio->Full,
        PlotStyle->{
            Red, 
            Sequence@@(
                Directive[
                    Thick, Opacity[.5], 
                    Dashed, ColorData[1][#]
                ]&/@Range[0, 5]
            )
        },
        ImageSize->Medium,
        PlotLegends->LineLegend[
            {"animate", Sequence@@(ToString@#&/@Range[0, 5])}
        ]
    ], Animator[Dynamic[t], 
                AnimationRunning->False, 
                AppearanceElements->{
                "PlayButton", "PauseButton", "ResetButton"}]},
        Alignment->Center           
    ]
]

Also, you can change the interpolation-function partsumtrans to get a different performance.