2 M2 Fourier Series answers in Mathematica
Note the function HeavisideTheta is 1 for x>0 and 0 for x<0 and is handy for
making the kind of function we need in a way mathematica knows how to integrate
I have plotted the Fourier series (at least afew terms) even though this was not asked for as this gives you an idea how
accurate the series is. Even if you don’t want to use Mathematica to check your integration please have a look at the graphs
Q1a
f = HeavisideTheta@-xD
HeavisideTheta@-xD
Plot@f, 8x, -Pi, Pi<D
1.0
0.8
0.6
0.4
0.2
-3
-2
-1
1
2
3
a0 = H1  PiL Integrate@f , 8x, -Pi, Pi<D
1
an = H1  PiL Integrate@f Cos@ n Pi x  PiD, 8x, -Pi, Pi<D
Sin@n ΠD
€€€€€€€€€€€€€€€€€€€€€€€
nΠ
Simplify@%, n Î IntegersD
0
bn = H1  PiL Integrate@f Sin@ n Pi x  PiD, 8x, -Pi, Pi<D
-1 + Cos@n ΠD
€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€
nΠ
Simplify@%, n Î IntegersD
n
-1 + H-1L
€€€€€€€€€€€€€€€€€€€€€€€€€€€
nΠ
2
FourierAnswers.nb
Plot@ a0  2 + Sum@ an Cos@n xD + bn Sin@n xD, 8n, 1, 20<D, 8x, -Pi, Pi<D
1.0
0.8
0.6
0.4
0.2
-3
-2
1
-1
2
3
Q1b
f = HeavisideTheta@xD HeavisideTheta@Pi  2 - xD
Π
HeavisideThetaB €€€ - xF HeavisideTheta@xD
2
Plot@f, 8x, -Pi, Pi<D
1.0
0.8
0.6
0.4
0.2
-3
-2
-1
1
2
3
a0 = H1  PiL Integrate@f , 8x, -Pi, Pi<D
1
€€€
2
an = H1  PiL Integrate@f Cos@ n Pi x  PiD, 8x, -Pi, Pi<D
nΠ
SinA €€€€€
€E
2
€€€€€€€€€€€€€€€€€€€€€€
nΠ
Simplify@%, n Î IntegersD
nΠ
SinA €€€€€
€E
2
€€€€€€€€€€€€€€€€€€€€€€
nΠ
bn = H1  PiL Integrate@f Sin@ n Pi x  PiD, 8x, -Pi, Pi<D
nΠ
2
2 SinA €€€€€
€E
4
€€€€€€€€€€€€€€€€€€€€€€€€€€€€€
nΠ
FourierAnswers.nb
Plot@ a0  2 + Sum@ an Cos@n xD + bn Sin@n xD, 8n, 1, 20<D, 8x, -Pi, Pi<D
1.0
0.8
0.6
0.4
0.2
-3
-2
1
-1
2
3
Q1c
f = 3 HeavisideTheta@xD
3 HeavisideTheta@xD
Plot@f, 8x, -5, 5<D
3.0
2.5
2.0
1.5
1.0
0.5
-4
-2
2
4
a0 = H1  5L Integrate@f , 8x, -5, 5<D
3
an = H1  5L Integrate@f Cos@ n Pi x  5D, 8x, -5, 5<D
3 Sin@n ΠD
€€€€€€€€€€€€€€€€€€€€€€€€€€€
nΠ
Simplify@%, n Î IntegersD
0
bn = H1  5L Integrate@f Sin@ n Pi x  5D, 8x, -5, 5<D
3 H-1 + Cos@n ΠDL
- €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€
nΠ
Simplify@%, n Î IntegersD
n
3 I-1 + H-1L M
- €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€
nΠ
3
4
FourierAnswers.nb
Plot@ a0  2 + Sum@ an Cos@n Pi x  5D + bn Sin@n Pi x  5D, 8n, 1, 20<D, 8x, -5, 5<D
3.0
2.5
2.0
1.5
1.0
0.5
-4
2
-2
4
Q1d
f=x
x
a0 = H1  H1  2LL Integrate@f , 8x, 0, 1<D
1
an = H1  H1  2LL Integrate@f Cos@ 2 n Pi xD, 8x, 0, 1<D
-1 + Cos@2 n ΠD + 2 n Π Sin@2 n ΠD
€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€
2 n2 Π2
Simplify@%, n Î IntegersD
0
bn = H1  H1  2LL Integrate@f Sin@ 2 n Pi xD, 8x, 0, 1<D
-2 n Π Cos@2 n ΠD + Sin@2 n ΠD
€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€
2 n2 Π2
Simplify@%, n Î IntegersD
1
- €€€€€€€€
nΠ
Plot@ a0  2 + Sum@ an Cos@2 n Pi xD + bn Sin@2 n Pi xD, 8n, 1, 20<D, 8x, 0, 1<D
1.0
0.8
0.6
0.4
0.2
0.2
Q1e
0.4
0.6
0.8
1.0
FourierAnswers.nb
f = x^2
x2
a0 = H1  H1  2LL Integrate@f , 8x, -1  2, 1  2<D
1
€€€
6
an = H1  H1  2LL Integrate@f Cos@ 2 n Pi xD, 8x, -1  2, 1  2<D
1
€€€€€€€€€€€€€€€€€ I2 n Π Cos@n ΠD + I-2 + n2 Π2 M Sin@n ΠDM
2 n3 Π3
Simplify@%, n Î IntegersD
n
H-1L
€€€€€€€€€€€€€€€
n2 Π2
bn = H1  H1  2LL Integrate@f Sin@ 2 n Pi xD, 8x, -1  2, 1  2<D
0
Plot@ a0  2 + Sum@ an Cos@2 n Pi xD + bn Sin@2 n Pi xD, 8n, 1, 3<D, 8x, -1  2, 1  2<D
0.20
0.15
0.10
0.05
-0.4
0.2
-0.2
0.4
Plot@ a0  2 + Sum@ an Cos@2 n Pi xD + bn Sin@2 n Pi xD, 8n, 1, 10<D, 8x, -1, 1<D
0.20
0.15
0.10
0.05
-1.0
-0.5
Q1f
f = Abs@ Sin@ w tDD
Abs@Sin@t wDD
0.5
1.0
5
6
FourierAnswers.nb
a0 = H1  HPi  wLL Integrate@f , 8t, -Pi  w, Pi  w<D
1
€€€
Π
4
Π Π
w IfBw Î Reals, €€€ , IntegrateBAbs@Sin@t wDD, :t, - €€€ , €€€ >, Assumptions ® Im@wD < 0 ÈÈ Im@wD > 0FF
w
w w
Simplify@%, w > 0D
4
€€€
Π
an = H1  HPi  wLL Integrate@f Cos@ Pi n t  HPi  wLD, 8t, -Pi  w, Pi  w<D
1
2 H1 + Cos@n ΠDL
€€€ w IfBw Î Reals, €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ ,
Π
w - n2 w
Π Π
IntegrateBAbs@Sin@t wDD Cos@n t wD, :t, - €€€ , €€€ >, Assumptions ® Im@wD < 0 ÈÈ Im@wD > 0FF
w w
Simplify@%, w > 0D
2 H1 + Cos@n ΠDL
€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€
Π - n2 Π
bn = H1  HPi  wLL Integrate@f Sin@ Pi n t  HPi  wLD, 8t, -Pi  w, Pi  w<D
1
€€€ w IfBw Î Reals, 0,
Π
Π Π
IntegrateBAbs@Sin@t wDD Sin@n t wD, :t, - €€€ , €€€ >, Assumptions ® Im@wD < 0 ÈÈ Im@wD > 0FF
w w
Simplify@%, w > 0D
0
w=2
2
2
2 H1 + Cos@n ΠDL
PlotB €€€€ + SumB €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ Cos@n w xD , 8n, 2, 10<F, 8x, -Pi  2, Pi  2<F
Π
Π - n2 Π
1.0
0.8
0.6
0.4
0.2
-1.5
-1.0
Extra
f = Abs@xD
Abs@xD
-0.5
0.5
1.0
1.5
FourierAnswers.nb
a0 = H1  1L Integrate@f , 8x, -1, 1<D
1
an = H1  1L Integrate@f Cos@ Pi n x  H1LD, 8x, -1, 1<D
2 H-1 + Cos@n ΠD + n Π Sin@n ΠDL
€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€
n2 Π2
Plot@ a0  2 + Sum@ an Cos@n Pi xD , 8n, 1, 10<D, 8x, -1, 1<D
1.0
0.8
0.6
0.4
0.2
-1.0
0.5
-0.5
1.0
Plot@ 8a0  2 + Sum@ an Cos@n Pi xD, 8n, 1, 20<D,
Sum@ H2  PiL H-H-1L ^ n  nL Sin@n Pi xD , 8n, 1, 20<D<, 8x, -1, 1<D
1.0
0.5
-1.0
0.5
-0.5
-0.5
-1.0
Sum@N@1  n ^ 2D, 8n, 1, 10 000<D
1.64483
Sum@N@H-1L ^ 8n + 1<  nD, 8n, 1, 1000<D
80.692647<
Q2a
f = Sign@xD
Out[5]=
Sign@xD
1.0
7
8
FourierAnswers.nb
In[7]:=
Plot@f, 8x, -Pi, Pi<D
1.0
0.5
Out[7]=
-3
-2
1
-1
2
3
-0.5
-1.0
It is odd
In[9]:=
bn = H1  PiL Integrate@f Sin@ n xD, 8x, -Pi, Pi<D
Out[9]=
2 H-1 + Cos@n ΠDL
- €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€
nΠ
In[11]:=
Plot@ Sum@ bn Sin@n xD, 8n, 20<D, 8x, -Pi, Pi<D
1.0
0.5
Out[11]=
-3
-2
1
-1
-0.5
-1.0
Q2b
In[12]:=
Out[12]=
f = Abs@xD
Abs@xD
2
3
FourierAnswers.nb
In[14]:=
Plot@f, 8x, -Pi  2, Pi  2<D
1.5
1.0
Out[14]=
0.5
-1.5
-1.0
0.5
-0.5
1.0
1.5
It is even
In[29]:=
Out[29]=
In[32]:=
an = Simplify@H2  PiL Integrate@f Cos@ 2 n xD, 8x, -Pi  2, Pi  2<DD
-1 + Cos@n ΠD + n Π Sin@n ΠD
€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€
n2 Π
Plot@ Pi  4 + Sum@ an Cos@2 n xD, 8n, 30<D, 8x, -Pi, Pi<D
1.5
1.0
Out[32]=
0.5
-3
-2
-1
1
2
3
Q3
In[57]:=
Out[57]=
In[54]:=
Out[54]=
In[55]:=
Out[55]=
fp = x HeavisideTheta@xD + Hx + 1L HeavisideTheta@-xD
H1 + xL HeavisideTheta@-xD + x HeavisideTheta@xD
fs = x
x
fc = Abs@xD
Abs@xD
9
10
FourierAnswers.nb
In[58]:=
Plot@8fp, fs, fc<, 8x, -1, 1<D
1.0
0.5
Out[58]=
-1.0
0.5
-0.5
1.0
-0.5
-1.0
In[66]:=
Out[66]=
In[65]:=
bpn = Simplify@ H2  1L Integrate@ fp Sin@2 n Pi xD, 8x, 0, 1<D, Element@n, IntegersDD
1
- €€€€€€€€
nΠ
bsn = Simplify@ H2  2L Integrate@ fs Sin@ n Pi xD, 8x, -1, 1<D, Element@n, IntegersDD
n
Out[65]=
2 H-1L
- €€€€€€€€€€€€€€€€€€€€
nΠ
In[77]:=
acn = Simplify@ H2  2L Integrate@ fc Cos@ n Pi xD, 8x, -1, 1<D, Element@n, IntegersDD
n
Out[77]=
In[80]:=
2 I-1 + H-1L M
€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€
n2 Π2
Plot@ 81  2 + Sum@ bpn Sin@2 n Pi xD, 8n, 1, 20<D,
Sum@ bsn Sin@n Pi xD, 8n, 1, 20<D, 1  2 + Sum@acn Cos@n Pi xD, 8n, 1, 20<D< , 8x, -1, 1<D
1.0
0.5
Out[80]=
-1.0
0.5
-0.5
-0.5
-1.0
Q4
In[83]:=
f = HeavisideTheta@x - 1D
Out[83]=
HeavisideTheta@-1 + xD
1.0
FourierAnswers.nb
In[85]:=
Plot@f, 8x, 0, 2<D
1.0
0.8
0.6
Out[85]=
0.4
0.2
0.5
In[89]:=
Out[89]=
In[90]:=
1.0
1.5
2.0
fc = HeavisideTheta@x - 1D + HeavisideTheta@-1 - xD
HeavisideTheta@-1 - xD + HeavisideTheta@-1 + xD
Plot@fc, 8x, -2, 2<D
1.0
0.8
0.6
Out[90]=
0.4
0.2
-2
In[93]:=
1
-1
2
fs = HeavisideTheta@x - 1D - HeavisideTheta@-1 - xD ; Plot@fs, 8x, -2, 2<D
1.0
0.5
Out[93]=
-2
1
-1
2
-0.5
-1.0
In[94]:=
bpn = Simplify@ H2  2L Integrate@ f Sin@ n Pi xD, 8x, 0, 2<D, Element@n, IntegersDD
n
Out[94]=
In[96]:=
-1 + H-1L
€€€€€€€€€€€€€€€€€€€€€€€€€€€
nΠ
acn = Simplify@ H2  4L Integrate@ fc Cos@ n Pi x  2D, 8x, -2, 2<D, Element@n, IntegersDD
nΠ
Out[96]=
2 SinA €€€€€
€E
2
- €€€€€€€€€€€€€€€€€€€€€€€€€€€
nΠ
11
12
FourierAnswers.nb
In[97]:=
bsn = Simplify@ H2  4L Integrate@ fs Sin@ n Pi x  2D, 8x, -2, 2<D, Element@n, IntegersDD
n
Out[97]=
In[109]:=
nΠ
2 I-H-1L + CosA €€€€€
€ EM
2
€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€
nΠ
Plot@8 1  2 + Sum@bpn Sin@n Pi xD, 8n, 1, 10<D,
1  2 + Sum@acn Cos@n Pi x  2D , 8n, 1, 10<D, Sum@bsn Sin@n Pi x  2D , 8n, 1, 10<D<, 8x, -2, 2<D
1.0
0.5
Out[109]=
-2
1
-1
2
-0.5
-1.0
Q5
In[36]:=
Out[36]=
In[37]:=
Out[37]=
In[38]:=
Out[38]=
In[50]:=
Out[50]=
In[40]:=
Out[40]=
2 H-1 + Cos@n ΠDL
- €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ . n ® 81, 2, 3, 4, 5, 6<
nΠ
4
4
4
: €€€ , 0, €€€€€€€€ , 0, €€€€€€€€ , 0>
Π
3Π
5Π
Integrate@Sign@xD ^ 2, 8x, -Pi, Pi<D
2Π
2 H-1 + Cos@n ΠDL
- €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ HPi  4L . n ® 81, 2, 3, 4, 5, 6<
nΠ
1
1
:1, 0, €€€ , 0, €€€ , 0>
3
5
Integrate@HPi Sign@xD  4L ^ 2, 8x, -Pi, Pi<D  Pi
Π2
€€€€€€
8
i 2 H-1 + Cos@n ΠDL
y
j- €€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ HPi  4Lz
z
SumBj
j
z ^ 2, 8n, 1, Infinity<F
nΠ
k
{
Π2
€€€€€€
8
Q6
n
In[41]:=
H-1L
an = €€€€€€€€€€€€€€€
n2 Π2
n
Out[41]=
In[46]:=
Out[46]=
H-1L
€€€€€€€€€€€€€€€
n2 Π2
Integrate@x ^ 2, 8x, -1  2, 1  2<D
1
€€€€€€
12
FourierAnswers.nb
In[49]:=
Out[49]=
In[44]:=
Out[44]=
Integrate@ H Pi ^ 2 Hx ^ 2 - 1  12LL ^ 2, 8x, -1  2, 1  2<D  H1  2L
Π4
€€€€€€
90
Sum@ 1  n ^ 4, 8n, 1, Infinity<D
Π4
€€€€€€
90
13