Lösungen

1

a

T=1;

b

f[t_]:=t-Floor[t];
Plot[f[t],{t,-3,3},AspectRatio->Automatic];

c   t' = tt

tt=t 2 Pi;
f1[tt_]:=f[tt/(2 Pi)];
f1[tt]

d

Plot[f1[tt],{tt,-7,7}];
Plot[f1[tt],{tt,-7,7},AspectRatio->Automatic];

2

a

s[n_,t_]:= 2 Sum[(-1)^(k+1) /k Sin[k t],{k,1,n}];
Table[Plot[s[n,t],{t,-2Pi,2Pi}],{n,1,6}];

b1

s[n_,t_]:= 2 Sum[(-1)^(k^2+1) /k^2 Sin[k^2 t],{k,1,n}];
Table[Plot[s[n,t],{t,-2Pi,2Pi}],{n,1,6}];

b2

s[n_,t_]:= 2 Sum[(-1)^(k+1) /k^2 Sin[k t],{k,1,n}];
Table[Plot[s[n,t],{t,-2Pi,2Pi}],{n,1,6}];

Plot[s[50,t],{t,-2Pi,2Pi}];

c

s[n_,t_]:= 2 Sum[(-1)^k /k Cos[k t],{k,1,n}];
Table[Plot[s[n,t],{t,-2Pi,2Pi}],{n,1,6}];

Plot[s[50,t],{t,-2Pi,2Pi}];

d1

s[n_,t_]:= 2 Sum[(-1)^(k^(1/2)+1) /(k^(1/2)) Sin[k^(1/2) t],{k,1,n}];
Table[Plot[s[n,t],{t,-2Pi,2Pi}],{n,1,2}];

d2

s[n_,t_]:= 2 Sum[(-1)^(k+1) /(k^(1/2)) Sin[k t],{k,1,n}];
Table[Plot[s[n,t],{t,-2Pi,2Pi}],{n,1,6}];

Plot[s[50,t],{t,-2Pi,2Pi}];

3

a

f[t_]:=2Pi(t-Floor[t]);
Plot[f[t],{t,-3,3},AspectRatio->Automatic];

tt=t 2 Pi;
f1[tt_]:=f[tt/(2 Pi)];
Plot[f1[tt],{tt,-7,7},AspectRatio->Automatic];

ttt=tt/(2Pi);
f2[ttt_]:=f1[ttt 2 Pi];
Plot[f2[ttt],{ttt,-3,3},AspectRatio->Automatic];

Remove["Global`*"]

<<Calculus`FourierTransform`

?*FourierTrig*

?FourierTrigSeries

f[t_]:=2Pi(t-Floor[t]);
Plot[f[t],{t,0,3}];

FourierTrigSeries[f[t], t, 10]

f1g10[tt_]:=FourierTrigSeries[f[t], t, 10]/.t->tt/(2 Pi);
Plot[Evaluate[f1g10[tt]],{tt,-7,7},AspectRatio->Automatic];

b

?FourierTrigSeries

Remove["Global`*"]

<<Calculus`FourierTransform`;
f[t_]:=2Pi(t-Floor[t])-Pi;
f1[t_]:=f[t/(2Pi)-1/2];
p1=Plot[f[t],{t,0,1}];
p2=Plot[f1[t],{t,-Pi,3Pi}];
p3=Plot[f1[t],{t,-Pi,3Pi},AspectRatio->Automatic,DisplayFunction->Identity,PlotStyle->{Text["Ueberlagerung",{1.5,-1},{0,0}]}];
f1g10[tt_]:=FourierTrigSeries[f[t], t, 10]/.t->tt/(2 Pi)-1/2;
p4=Plot[Evaluate[f1g10[tt]],{tt,-Pi,Pi},AspectRatio->Automatic,DisplayFunction->Identity];
Show[p3,p4,DisplayFunction->$DisplayFunction];
FourierTrigSeries[f[t], t, 10];

c

Remove["Global`*"]

<<Calculus`FourierTransform`;
f[t_]:=t^2;
f1[t_]:=f[t 2 Pi];
p1=Plot[f[t],{t,-Pi,Pi},AspectRatio->Automatic];
p2=Plot[f1[t],{t,-0.5,0.5}];
Plot[Evaluate[FourierTrigSeries[f1[t], t, 10]],{t,-0.5,0.5}];
f1g10[tt_]:=FourierTrigSeries[f1[t], t, 10]/.t->tt/(2 Pi);
p3=Plot[Evaluate[f1g10[tt]],{tt,-Pi,Pi},AspectRatio->Automatic,DisplayFunction->Identity,PlotStyle->{Text["Ueberlagerung",{1.5,-1},{0,0}]}];
Show[p1,p3,DisplayFunction->$DisplayFunction];
FourierTrigSeries[f1[t], t, 10];

d

Remove["Global`*"]

<<Calculus`FourierTransform`;
f[t_]:=Sin[t];
f1[t_]:=f[t 2 Pi];
p1=Plot[f[t],{t,-Pi,Pi},AspectRatio->Automatic];
p2=Plot[f1[t],{t,-0.5,0.5}];
Plot[Evaluate[FourierTrigSeries[f1[t], t, 10]],{t,-0.5,0.5}];
f1g10[tt_]:=FourierTrigSeries[f1[t], t, 10]/.t->tt/(2 Pi);
p3=Plot[Evaluate[f1g10[tt]],{tt,-Pi,Pi},AspectRatio->Automatic,DisplayFunction->Identity,PlotStyle->{Text["Ueberlagerung",{1.5,-1},{0,0}]}];
Show[p1,p3,DisplayFunction->$DisplayFunction];
FourierTrigSeries[f1[t], t, 10];

e

Remove["Global`*"]

<<Calculus`FourierTransform`;
f[t_]:=Sin[t+1];
f1[t_]:=f[t 2 Pi];
p1=Plot[f[t],{t,-Pi,Pi},AspectRatio->Automatic];
p2=Plot[f1[t],{t,-0.5,0.5}];
Plot[Evaluate[FourierTrigSeries[f1[t], t, 10]],{t,-0.5,0.5}];
f1g10[tt_]:=FourierTrigSeries[f1[t], t, 10]/.t->tt/(2 Pi);
p3=Plot[Evaluate[f1g10[tt]],{tt,-Pi,Pi},AspectRatio->Automatic,DisplayFunction->Identity,PlotStyle->{Text["Ueberlagerung",{1.5,-1},{0,0}]}];
Show[p1,p3,DisplayFunction->$DisplayFunction];
FourierTrigSeries[f1[t], t, 10];

f

Remove["Global`*"]

<<Calculus`FourierTransform`;
f[t_]:=Sin[t]^2;
f1[t_]:=f[t 2 Pi];
p1=Plot[f[t],{t,-Pi,Pi},AspectRatio->Automatic];
p2=Plot[f1[t],{t,-0.5,0.5}];
Plot[Evaluate[FourierTrigSeries[f1[t], t, 10]],{t,-0.5,0.5}];
f1g10[tt_]:=FourierTrigSeries[f1[t], t, 10]/.t->tt/(2 Pi);
p3=Plot[Evaluate[f1g10[tt]],{tt,-Pi,Pi},AspectRatio->Automatic,DisplayFunction->Identity,PlotStyle->{Text["Ueberlagerung",{1.5,-1},{0,0}]}];
Show[p1,p3,DisplayFunction->$DisplayFunction];
FourierTrigSeries[f1[t], t, 10];

g

Remove["Global`*"]

<<Calculus`FourierTransform`;
f[t_]:=Cos[t]^2;
f1[t_]:=f[t 2 Pi];
p1=Plot[f[t],{t,-Pi,Pi},AspectRatio->Automatic];
p2=Plot[f1[t],{t,-0.5,0.5}];
Plot[Evaluate[FourierTrigSeries[f1[t], t, 10]],{t,-0.5,0.5}];
f1g10[tt_]:=FourierTrigSeries[f1[t], t, 10]/.t->tt/(2 Pi);
p3=Plot[Evaluate[f1g10[tt]],{tt,-Pi,Pi},AspectRatio->Automatic,DisplayFunction->Identity,PlotStyle->{Text["Ueberlagerung",{1.5,-1},{0,0}]}];
Show[p1,p3,DisplayFunction->$DisplayFunction];
FourierTrigSeries[f1[t], t, 10];

h

Remove["Global`*"]

<<Calculus`FourierTransform`;
f[t_]:=E^t;
f1[t_]:=f[t 2 Pi];
p1=Plot[f[t],{t,-Pi,Pi},AspectRatio->Automatic,PlotRange->{0,10}];
p2=Plot[f1[t],{t,-0.5,0.5}];
Plot[Evaluate[FourierTrigSeries[f1[t], t, 10]],{t,-0.5,0.5}];
f1g10[tt_]:=FourierTrigSeries[f1[t], t, 10]/.t->tt/(2 Pi);
p3=Plot[Evaluate[f1g10[tt]],{tt,-Pi,Pi},AspectRatio->Automatic,DisplayFunction->Identity,PlotStyle->{Text["Ueberlagerung",{1.5,-1},{0,0}]},PlotRange->{0,10}];
Show[p1,p3,DisplayFunction->$DisplayFunction];
FourierTrigSeries[f1[t], t, 10];

i

Remove["Global`*"]

<<Calculus`FourierTransform`;
f[t_]:=Cosh[t];
f1[t_]:=f[t 2 Pi];
p1=Plot[f[t],{t,-Pi,Pi},AspectRatio->Automatic,PlotRange->{0,10}];
p2=Plot[f1[t],{t,-0.5,0.5}];
Plot[Evaluate[FourierTrigSeries[f1[t], t, 10]],{t,-0.5,0.5}];
f1g10[tt_]:=FourierTrigSeries[f1[t], t, 10]/.t->tt/(2 Pi);
p3=Plot[Evaluate[f1g10[tt]],{tt,-Pi,Pi},AspectRatio->Automatic,DisplayFunction->Identity,PlotStyle->{Text["Ueberlagerung",{1.5,-1},{0,0}]},PlotRange->{0,10}];
Show[p1,p3,DisplayFunction->$DisplayFunction];
FourierTrigSeries[f1[t], t, 10];