f(t) =t^2 -3<t<3​
​a0

In[]:=
1/(2*Pi)*Integrate[t^2,{t,-3,3}]
Out[]=
9
π

an

In[]:=
1/(2*Pi)*Integrate[t^2*Cos[n*t],{t,-3,3}]
Out[]=
6nCos[3n]+(-2+9
2
n
)Sin[3n]
3
n
π
​

bn

In[]:=
1/(2*Pi)*Integrate[t^2*Sin[n*t],{t,-3,3}]
Out[]=
0

Eerie d fourier

In[]:=
a0=
9
π
​​a[n_]=
6nCos[3n]+(-2+9
2
n
)Sin[3n]
3
n
π
​​
​​f[t_,Nmax_]:=a0+Sum[a[n]*Cos[n*t],{n,1,Nmax}]​​f[t,20]​​Plot[{t^2,f[t,20]},{t,1,-3,3}];​​Manipulate[Plot[{t^2,f[t,m]},{t,-3,3}],{m,1,10}]
Out[]=
9
π
Out[]=
6nCos[3n]+(-2+9
2
n
)Sin[3n]
3
n
π
Out[]=
Out[]=
9
π
+
Cos[t](6Cos[3]+7Sin[3])
π
+
Cos[2t](12Cos[6]+34Sin[6])
8π
+
Cos[3t](18Cos[9]+79Sin[9])
27π
+
Cos[4t](24Cos[12]+142Sin[12])
64π
+
Cos[5t](30Cos[15]+223Sin[15])
125π
+
Cos[6t](36Cos[18]+322Sin[18])
216π
+
Cos[7t](42Cos[21]+439Sin[21])
343π
+
Cos[8t](48Cos[24]+574Sin[24])
512π
+
Cos[9t](54Cos[27]+727Sin[27])
729π
+
Cos[10t](60Cos[30]+898Sin[30])
1000π
+
Cos[11t](66Cos[33]+1087Sin[33])
1331π
+
Cos[12t](72Cos[36]+1294Sin[36])
1728π
+
Cos[13t](78Cos[39]+1519Sin[39])
2197π
+
Cos[14t](84Cos[42]+1762Sin[42])
2744π
+
Cos[15t](90Cos[45]+2023Sin[45])
3375π
+
Cos[16t](96Cos[48]+2302Sin[48])
4096π
+
Cos[17t](102Cos[51]+2599Sin[51])
4913π
+
Cos[18t](108Cos[54]+2914Sin[54])
5832π
+
Cos[19t](114Cos[57]+3247Sin[57])
6859π
+
Cos[20t](120Cos[60]+3598Sin[60])
8000π
Plot
:Range specification {t,1,-3,3} is not of the form {x, xmin, xmax}.
Out[]=
​
m
-3
-2
-1
1
2
3
2
4
6
8

forma exponencial

In[]:=
1/(2*Pi)*Integrate[t^2*Exp[I*k*t],{t,-3,3}]
Out[]=
12kCos[3k]+2(-2+9
2
k
)Sin[3k]
2
3
k
π
c[k_]=
12kCos[3k]+2(-2+9
2
k
)Sin[3k]
2
3
k
π
​​f[n_]:=Sum[c[k]*Exp[I*k*t],{t,-3,3}]

AQUI ES DONDE TENGO PROBLEMAS

Out[]=
12kCos[3k]+2(-2+9
2
k
)Sin[3k]
2
3
k
π
Plot
:Plot called with 1 argument; 2 arguments are expected.
Out[]=
Plot[f[t,10]]