In[]:=
f[x_]=Which[-1<x<0,-1,0<x<1,1]
Out[]=
Which[-1<x<0,-1,0<x<1,1]
In[]:=
Plot[f[x],{x,-1,1}]
Out[]=
In[]:=
a[n_]:=(2/L)*Integrate[f[x]*Cos[2n*Pi*x/L],{x,-L/2,L/2}]
In[]:=
a[0]:=(1/L)*Integrate[f[x],{x,-L/2,L/2}]
In[]:=
b[n_]:=(2/L)*Integrate[f[x]*Sin[2n*Pi*x/L],{x,-L/2,L/2}]
In[]:=
F[x_,N_]:=a[0]+Sum[a[n]*Cos[2n*Pi*x/L]+b[n]*Sin[2n*Pi*x/L],{n,1,N}]
In[]:=
p[N_,a_]:=Plot[Evaluate[F[x,N]],{x,-a,a},PlotRange->All,PlotPoints->200]
In[]:=
L=2;
In[]:=
f[x_]=If[x>0,1,-1];
In[]:=
a[n]
Out[]=
0
In[]:=
a[0]
Out[]=
0
In[]:=
b[n]
Out[]=
2-2Cos[nπ]
nπ
In[]:=
Simplify[%,n∈Integers]
Out[]=
-
2(-1+)
n
(-1)
nπ
In[]:=
Table[F[0.5,k],{k,0,20}]
Out[]=
{0,1.27324,1.27324,0.848826,0.848826,1.10347,1.10347,0.921583,0.921583,1.06305,1.06305,0.947305,0.947305,1.04525,1.04525,0.960364,0.960364,1.03526,1.03526,0.968248,0.968248}
In[]:=
Integrate[f[x]^2,{x,-L,L}]
Out[]=
4
In[]:=
(a[0]^2)/2+Sum[(a[n]^2+b[n]^2),{n,1,Infinity}]<=Integrate[f[x]^2,{x,-L,L}]
Out[]=
True