Absolute Value of x from -Pi to Pi.
Absolute Value of x from -Pi to Pi.
part1=Plot[Abs[x+2Pi],{x,-3Pi,-Pi},DisplayFunction->Identity];
part2=Plot[Abs[x],{x,-Pi,Pi},DisplayFunction->Identity];
part3=Plot[Abs[x-2Pi],{x,Pi,3Pi},DisplayFunction->Identity];
absgraph=Show[part1,part2,part3,
DisplayFunction->$DisplayFunction,AspectRatio->Automatic];
In[2]:=
Clear[a0,a,b,c,Sn]
a0=(1/(2Pi))(Integrate[-x,{x,-Pi,0}]+Integrate[x,{x,0,Pi}]);
a0
Out[2]=
Pi -- 2
In[3]:=
a[n_]:=a[n]=(1/Pi)(Integrate[-x Cos[n x],{x,-Pi,0}]+
Integrate[x Cos[n x],{x,0,Pi}]);
Table[a[n],{n,1,5}]
Out[3]=
-4 -4 -4
{--, 0, ----, 0, -----}
Pi 9 Pi 25 Pi
In[4]:=
b[n_]:=b[n]=(1/Pi)(Integrate[-x Sin[n x],{x,-Pi,0}]+
Integrate[x Sin[n x],{x,0,Pi}]);
Table[b[n],{n,1,5}]
Out[4]=
{0, 0, 0, 0, 0}
No surprise there, I hope.
In[5]:=
Sn=a0+Sum[a[n]Cos[n x],{n,1,5}]
Out[5]=
Pi 4 Cos[x] 4 Cos[3 x] 4 Cos[5 x] -- - -------- - ---------- - ---------- 2 Pi 9 Pi 25 Pi
In[6]:=
p1=Plot[Sn,{x,-Pi,Pi}];
In[7]:=
Show[absgraph,p1];
In[8]:=
Clear[c]
In[9]:=
c[n_]:=c[n]=(1/(2 Pi))(Integrate[-x Exp[- I n x],{x,-Pi,0}]+
Integrate[x Exp[- I n x],{x,0,Pi}]);
Table[c[n],{n,-5,5}]
Out[9]=
2 -1 + 5 I Pi I
-(--) + ----------- - -- (-I + 5 Pi)
25 25 25
{------------------------------------,
2 Pi
1 I 1 - 4 I Pi
-(-) - -- (I - 4 Pi) + ----------
8 16 16
---------------------------------,
2 Pi
2 -1 + 3 I Pi I
-(-) + ----------- - - (-I + 3 Pi)
9 9 9
----------------------------------,
2 Pi
1 I 1 - 2 I Pi
-(-) - - (I - 2 Pi) + ----------
2 4 4
--------------------------------,
2 Pi
-3 + I Pi - I (-I + Pi) Pi -3 - I Pi + I (I + Pi)
-----------------------, --, ----------------------,
2 Pi 2 2 Pi
1 I 1 + 2 I Pi
-(-) + - (-I - 2 Pi) + ----------
2 4 4
---------------------------------,
2 Pi
2 -1 - 3 I Pi I
-(-) + ----------- + - (I + 3 Pi)
9 9 9
---------------------------------,
2 Pi
1 I 1 + 4 I Pi
-(-) + -- (-I - 4 Pi) + ----------
8 16 16
----------------------------------,
2 Pi
2 -1 - 5 I Pi I
-(--) + ----------- + -- (I + 5 Pi)
25 25 25
-----------------------------------}
2 Pi
In[10]:=
ISn=Sum[c[n]Exp[I n x],{n,-5,5}]
Out[10]=
-2 I x 1 I 1 - 2 I Pi
E (-(-) - - (I - 2 Pi) + ----------)
2 4 4
------------------------------------------ +
2 Pi
2 I x 1 I 1 + 2 I Pi
E (-(-) + - (-I - 2 Pi) + ----------)
2 4 4
------------------------------------------ +
2 Pi
-4 I x 1 I 1 - 4 I Pi
E (-(-) - -- (I - 4 Pi) + ----------)
8 16 16
------------------------------------------- +
2 Pi
4 I x 1 I 1 + 4 I Pi
E (-(-) + -- (-I - 4 Pi) + ----------)
8 16 16 Pi
------------------------------------------- + -- +
2 Pi 2
-I x
E (-3 + I Pi - I (-I + Pi))
------------------------------- +
2 Pi
I x
E (-3 - I Pi + I (I + Pi))
----------------------------- +
2 Pi
-3 I x 2 -1 + 3 I Pi I
E (-(-) + ----------- - - (-I + 3 Pi))
9 9 9
-------------------------------------------- +
2 Pi
3 I x 2 -1 - 3 I Pi I
E (-(-) + ----------- + - (I + 3 Pi))
9 9 9
------------------------------------------ +
2 Pi
-5 I x 2 -1 + 5 I Pi I
E (-(--) + ----------- - -- (-I + 5 Pi))
25 25 25
---------------------------------------------- +
2 Pi
5 I x 2 -1 - 5 I Pi I
E (-(--) + ----------- + -- (I + 5 Pi))
25 25 25
--------------------------------------------
2 Pi
In[11]:=
ComplexExpand[ISn]
Out[11]=
Pi 4 Cos[x] 4 Cos[3 x] 4 Cos[5 x] -- - -------- - ---------- - ---------- 2 Pi 9 Pi 25 Pi
Our partial sums are indeed the same!
In[12]:=
Simplify[Sn-ISn]
Out[12]=
0
In[13]:=
p2=Plot[ISn,{x,-Pi,Pi}];
