Ramp-Like Function
Ramp-Like Function
part1=Plot[0,{x,-2,0},DisplayFunction->Identity];
part2=Plot[2-x,{x,0,2},DisplayFunction->Identity];
graph=Show[part1,part2,
DisplayFunction->$DisplayFunction,AspectRatio->Automatic];
In[15]:=
Clear[a0,a,b,c,Sn]
a0=(1/4)(Integrate[0,{x,-2,0}]+Integrate[2-x,{x,0,2}]);
a0
Out[15]=
1 - 2
In[16]:=
a[n_]:=a[n]=(1/2)(Integrate[0 Cos[n Pi x/2],{x,-2,0}]+
Integrate[(2-x) Cos[n Pi x/2],{x,0,2}]);
Table[a[n],{n,1,5}]
Out[16]=
4 4 4
{---, 0, -----, 0, ------}
2 2 2
Pi 9 Pi 25 Pi
In[17]:=
b[n_]:=b[n]=(1/2)(Integrate[0 Sin[n Pi x/2],{x,-2,0}]+
Integrate[(2-x) Sin[n Pi x/2],{x,0,2}]);
Table[b[n],{n,1,5}]
Out[17]=
2 1 2 1 2
{--, --, ----, ----, ----}
Pi Pi 3 Pi 2 Pi 5 Pi
In[18]:=
Sn=a0+Sum[a[n]Cos[n Pi x/2]+b[n]Sin[n Pi x/2],{n,1,5}]
Out[18]=
Pi x 3 Pi x 5 Pi x
4 Cos[----] 4 Cos[------] 4 Cos[------]
1 2 2 2
- + ----------- + ------------- + ------------- +
2 2 2 2
Pi 9 Pi 25 Pi
Pi x 3 Pi x
2 Sin[----] 2 Sin[------]
2 Sin[Pi x] 2 Sin[2 Pi x]
----------- + --------- + ------------- + ----------- +
Pi Pi 3 Pi 2 Pi
5 Pi x
2 Sin[------]
2
-------------
5 Pi
In[19]:=
p3=Plot[Sn,{x,-2,2}];
In[20]:=
Show[graph,p3];
In[21]:=
Clear[c]
In[22]:=
c[n_]:=c[n]=(1/4)(Integrate[0 Exp[- I n Pi x/2],{x,-2,0}]+
Integrate[(2-x) Exp[- I n Pi x/2],{x,0,2}]);
Table[c[n],{n,-5,5}]
Out[22]=
2 I I
--- (2 I - 10 Pi) - (I - 4 Pi)
4 25 -1 4
------ - ----------------- ----- - ------------
2 2 2 2
25 Pi Pi 4 Pi Pi
{--------------------------, --------------------,
4 4
2 I
--- (2 I - 6 Pi)
4 9 -2 I (I - 2 Pi)
----- - ---------------- -Pi - ------------
2 2 2
9 Pi Pi Pi
------------------------, --------------------,
4 4
4 2 I (2 I - 2 Pi) 4 2 I (-2 I - 2 Pi)
--- - ---------------- --- + -----------------
2 2 2 2
Pi Pi 1 Pi Pi
----------------------, -, -----------------------,
4 2 4
2 I
--- (-2 I - 6 Pi)
-2 I (-I - 2 Pi) 4 9
-Pi + ------------- ----- + -----------------
2 2 2
Pi 9 Pi Pi
---------------------, -------------------------,
4 4
I 2 I
- (-I - 4 Pi) --- (-2 I - 10 Pi)
-1 4 4 25
----- + ------------- ------ + ------------------
2 2 2 2
4 Pi Pi 25 Pi Pi
---------------------, ---------------------------}
4 4
In[23]:=
ISn=Sum[c[n]Exp[I n Pi x/2],{n,-5,5}];
In[24]:=
ComplexExpand[ISn]
Out[24]=
Pi x 3 Pi x 5 Pi x
4 Cos[----] 4 Cos[------] 4 Cos[------]
1 2 2 2
- + ----------- + ------------- + ------------- +
2 2 2 2
Pi 9 Pi 25 Pi
Pi x 3 Pi x
2 Sin[----] 2 Sin[------]
2 Sin[Pi x] 2 Sin[2 Pi x]
----------- + --------- + ------------- + ----------- +
Pi Pi 3 Pi 2 Pi
5 Pi x
2 Sin[------]
2
-------------
5 Pi
In[25]:=
p4=Plot[ISn,{x,-2,2}];
In[26]:=
Show[graph,p4];
In[27]:=
Sn
ComplexExpand[ISn]
Out[27]=
Pi x 3 Pi x 5 Pi x
4 Cos[----] 4 Cos[------] 4 Cos[------]
1 2 2 2
- + ----------- + ------------- + ------------- +
2 2 2 2
Pi 9 Pi 25 Pi
Pi x 3 Pi x
2 Sin[----] 2 Sin[------]
2 Sin[Pi x] 2 Sin[2 Pi x]
----------- + --------- + ------------- + ----------- +
Pi Pi 3 Pi 2 Pi
5 Pi x
2 Sin[------]
2
-------------
5 Pi
Out[28]=
Pi x 3 Pi x 5 Pi x
4 Cos[----] 4 Cos[------] 4 Cos[------]
1 2 2 2
- + ----------- + ------------- + ------------- +
2 2 2 2
Pi 9 Pi 25 Pi
Pi x 3 Pi x
2 Sin[----] 2 Sin[------]
2 Sin[Pi x] 2 Sin[2 Pi x]
----------- + --------- + ------------- + ----------- +
Pi Pi 3 Pi 2 Pi
5 Pi x
2 Sin[------]
2
-------------
5 Pi
In[29]:=
Simplify[Sn-ISn]
Out[29]=
0