#
#
#
# WORKSHEET # 21
#
# Some Fourier Series
#
#
> restart;
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# For a function f(x) defined on the interval [0,2*Pi] the Fourier series
# is a linear combination of the periodic functions cos(n*x), sin(n*x)
# and a constant term.
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> f(x)=(1/2)*a[0]+Sum(a[n]*cos(n*x)+b[n]*sin(n*x),n=1..infinity);
/infinity \
| ----- |
| \ |
f(x) = 1/2 a[0] + | ) (a[n] cos(n x) + b[n] sin(n x))|
| / |
| ----- |
\ n = 1 /
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# The coefficients in this formula are given by the following formulas.
# The formula for a[n] is valid for n=0,1,... and that for b[n] is valid for
# n=1,2,...
--------------------------------------------------------------------------------
> a[n]=(1/Pi)*Int(f(x)*cos(n*x),x=0..2*Pi);
2 Pi
/
|
| f(x) cos(n x) dx
|
/
0
a[n] = ----------------------
Pi
--------------------------------------------------------------------------------
> b[n]=(1/Pi)*Int(f(x)*sin(n*x),x=0..2*Pi);
2 Pi
/
|
| f(x) sin(n x) dx
|
/
0
b[n] = ----------------------
Pi
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# If we assume that f(x) is actually represented by the infinite series
# above and we can integrate term-by-term in this series then the
# formulas for the coefficients follow from the following orthogonal
# properties of sin(x) and cos(x):
#
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> weknow:=cos(2*m*Pi)=1,sin(2*m*Pi)=0,cos(2*n*Pi)=1,sin(2*n*Pi)=0;\
weknow := cos(2 m Pi) = 1, sin(2 m Pi) = 0, cos(2 n Pi) = 1, sin(2 n Pi) = 0
--------------------------------------------------------------------------------
> subs(weknow,Int(sin(m*x)*sin(m*x),x=0..2*Pi)=int(sin(m*x)*sin(m*x),x=0..2*Pi));
2 Pi
/
| 2
| sin(m x) dx = Pi
|
/
0
--------------------------------------------------------------------------------
> subs(weknow,Int(cos(m*x)*cos(m*x),x=0..2*Pi)=int(cos(m*x)*cos(m*x),x=0..2*Pi));
2 Pi
/
| 2
| cos(m x) dx = Pi
|
/
0
--------------------------------------------------------------------------------
> Int(sin(m*x)*cos(n*x),x=0..2*Pi)=0;
2 Pi
/
|
| sin(m x) cos(n x) dx = 0
|
/
0
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# All other integrals involving sin(m*x) and cos(n*x) are zero. That is
# if m does not equal n then:
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> Int(sin(m*x)*sin(n*x),x=0..2*Pi)=0,Int(cos(m*x)*cos(n*x),x=0..2*Pi)=0;
2 Pi 2 Pi
/ /
| |
| sin(m x) sin(n x) dx = 0, | cos(m x) cos(n x) dx = 0
| |
/ /
0 0
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# Now we will compute the Fourier series for some simple functions.
--------------------------------------------------------------------------------
> f:=x*(2*Pi-x);
f := x (2 Pi - x)
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> F:=plot(f,x=0..2*Pi):
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> with(plots):
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> display(F,title=`f(x)=x*(2*Pi-x)`);
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#
** Maple V Graphics **
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> B:=int(f*sin(n*x),x=0..2*Pi);
n sin(2 n Pi) Pi + cos(2 n Pi) 2
B := - 2 ------------------------------ + ----
3 3
n n
--------------------------------------------------------------------------------
> subs(weknow,B);
0
--------------------------------------------------------------------------------
# Thus all the sine coefficients are zero.
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> A:=int(f*cos(n*x),x=0..2*Pi);
- n cos(2 n Pi) Pi + sin(2 n Pi) Pi
A := 2 -------------------------------- - 2 ----
3 2
n n
--------------------------------------------------------------------------------
> A:=subs(weknow,A);
Pi
A := - 4 ----
2
n
--------------------------------------------------------------------------------
> a0:=int(f,x=0..2*Pi)/(2*Pi);
2
a0 := 2/3 Pi
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> a:=unapply(A/Pi,n);
4
a := n -> - ----
2
n
--------------------------------------------------------------------------------
> a(5);
-4/25
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> S:=n->a0+sum(a(k)*cos(k*x),k=1..n);
/ n \
|----- |
| \ |
S := n -> a0 + | ) a(k) cos(k x)|
| / |
|----- |
\k = 1 /
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> S(1);
2
2/3 Pi - 4 cos(x)
--------------------------------------------------------------------------------
> S(2);
2
2/3 Pi - 4 cos(x) - cos(2 x)
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> S(3);
2
2/3 Pi - 4 cos(x) - cos(2 x) - 4/9 cos(3 x)
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> sun.rise;
sunrise
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> subscript.10;
subscript10
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> for j from 1 to 5 do S(j) od;j:='j':
2
2/3 Pi - 4 cos(x)
2
2/3 Pi - 4 cos(x) - cos(2 x)
2
2/3 Pi - 4 cos(x) - cos(2 x) - 4/9 cos(3 x)
2
2/3 Pi - 4 cos(x) - cos(2 x) - 4/9 cos(3 x) - 1/4 cos(4 x)
2
2/3 Pi - 4 cos(x) - cos(2 x) - 4/9 cos(3 x) - 1/4 cos(4 x) - 4/25 cos(5 x)
--------------------------------------------------------------------------------
> for j from 1 to 10 do P.j:=plot(S(j),x=0..2*Pi) od:
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> F:=plot(f,x=0..2*Pi,style=POINT):
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> display(F,title=`f(x)=x*(2Pi-x)`);
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#
** Maple V Graphics **
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> display([F,P1],title=`first partial sum`);
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#
** Maple V Graphics **
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> display([F,P2],title=`second partial sum`);
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#
** Maple V Graphics **
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> display([F,P3],title=`third partial sum`);\
display([F,P4],title=`fourth partial sum`);\
display([F,P3],title=`fifth partial sum`);
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#
** Maple V Graphics **
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#
** Maple V Graphics **
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#
** Maple V Graphics **
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> for j from 1 to 20 do print(j,evalf(subs(x=0,S(j)-f))) od;j:='j':
1, 2.579736270
2, 1.579736270
3, 1.135291826
4, .885291826
5, .725291826
6, .614180714
7, .532548061
8, .470048061
9, .420665345
10, .380665345
11, .347607494
12, .319829716
13, .296161077
14, .275752914
15, .257975136
16, .242350136
17, .228509306
18, .216163627
19, .205083294
20, .195083294
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> for j from 1 to 20 do print(j,evalf(subs(x=Pi,S(j)-f))) od;j:='j':
1, .710131866
2, -.289868134
3, .1545763104
4, -.0954236896
5, .0645763104
6, -.0465348007
7, .03509785236
8, -.02740214764
9, .02198056841
10, -.01801943159
11, .01503841965
12, -.01273935813
13, .01092928092
14, -.00947888235
15, .00829889543
16, -.00732610457
17, .00651472588
18, -.00583095313
19, .00524937928
20, -.00475062072
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> fourier:=a0+Sum(a(k)*cos(k*x),k=1..infinity);
/infinity \
| ----- |
2 | \ cos(k x)|
fourier := 2/3 Pi + | ) - 4 --------|
| / 2 |
| ----- k |
\ k = 1 /
--------------------------------------------------------------------------------
> subs(x=0,fourier);
/infinity \
| ----- |
2 | \ cos(0)|
2/3 Pi + | ) - 4 ------|
| / 2 |
| ----- k |
\ k = 1 /
--------------------------------------------------------------------------------
> eval(");
/infinity \
| ----- |
2 | \ 4 |
2/3 Pi + | ) - ----|
| / 2 |
| ----- k |
\ k = 1 /
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> evalf(");\
-8
.3*10
--------------------------------------------------------------------------------
> Sum(1/k^2,k=1..infinity)=sum(1/k^2,k=1..infinity);
infinity
-----
\ 1 2
) ---- = 1/6 Pi
/ 2
----- k
k = 1
--------------------------------------------------------------------------------
> mod2Pi:=x->x-2*Pi*floor(x/(2*Pi));
x
mod2Pi := x -> x - 2 Pi floor(1/2 ----)
Pi
--------------------------------------------------------------------------------
> fperiodic:=x->mod2Pi(x)*(2*Pi-mod2Pi(x));
fperiodic := x -> mod2Pi(x) (2 Pi - mod2Pi(x))
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> Fper:=plot(fperiodic(x),x=-4*Pi..4*Pi,style=POINT,numpoints=200):
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> display(Fper,title=`periodic extension of f(x)`);
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#
** Maple V Graphics **
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> for j from 1 to 5 do Pper.j:=plot(S(j),x=-4*Pi..4*Pi,numpoints=200) od:
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> display([Fper,Pper1],title=`first partial sum`);display([Fper,Pper2],title=`second partial sum`);\
display([Fper,Pper3],title=`third partial sum`);
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#
** Maple V Graphics **
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#
** Maple V Graphics **
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#
** Maple V Graphics **
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> fperiodic(Pi);
2
Pi
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> approx:=a0+sum(a(k)*cos(k*Pi),k=1..infinity);
/infinity \
| ----- |
2 | \ cos(k Pi)|
approx := 2/3 Pi + | ) - 4 ---------|
| / 2 |
| ----- k |
\ k = 1 /
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> wealsoknow:=cos(k*Pi)=(-1)^k;
k
wealsoknow := cos(k Pi) = (-1)
--------------------------------------------------------------------------------
> subs(wealsoknow,approx);
/infinity \
| ----- k|
2 | \ (-1) |
2/3 Pi + | ) - 4 -----|
| / 2 |
| ----- k |
\ k = 1 /
--------------------------------------------------------------------------------
> eval(");
2
2/3 Pi + 4 hypergeom([1, 1, 1], [2, 2], -1)
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> ?hypergeom
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> evalf("");
Error, (in evalf/Hypergeom) Too many iterations without convergence
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> odd:=sum(1/(2*k-1)^2,k=1.. infinity);
2
odd := 1/8 Pi
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> even:=sum(1/(2*k)^2,k=1.. infinity);
2
even := 1/24 Pi
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> odd-even;
2
1/12 Pi
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> a0+4*";
2
Pi
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