Fourier Series Expansion 

Yasuyuki Nakamura
Graduate School of Information Science, Nagoya University
A4-2(780), Furo-cho, Chikusa-ku, Nagoya, 464-8601, Japan
nakamura@nagoya-u.jp
http://www.phys.cs.is.nagoya-u.ac.jp/~nakamura/ 

Definition 

A periodic function with a period can be expressed as follows with sine functions and cosine functions: 

where and are Fourier coefficients and are defined as  

Example 

Here we consider the case of a function with a period 2 () for the simplicity.  

Function 

>

restart:

 

>	with(DocumentTools):

 

>

c := 0:  f1 := 0:  f2 := 1:  f := x -> piecewise(-1 <= x and x <= c, f1, c <= x and x <= 1, f2):  SetProperty(ComboBox0, value, "Step");  SetProperty(TextArea0, value, 0);  SetProperty(TextArea1, value, 0);  SetProperty(TextArea2, value, 0);  SetProperty(TextArea3, value, 1);  SetProperty(TextArea3, visible, true);  SetProperty(Plot0, value, plot(f(x), x=-1..1, thickness=2, scaling=constrained));  SetProperty(MathContainer0, value, NULL);  SetProperty(MathContainer1, value, NULL);  SetProperty(MathContainer2, value, NULL);

 

>

draw := proc()  global f, c, f1, f2;  c := parse(GetProperty(TextArea0, value));  f1 := parse(GetProperty(TextArea2, value));  f2 := parse(GetProperty(TextArea3, value));  f := x -> piecewise(-1 <= x and x <= c, f1, c <= x and x <= 1, f2);  SetProperty(Plot0, value, plot(f(x), x=-1..1, thickness=2, scaling=constrained));  SetProperty(MathContainer0, value, NULL);  SetProperty(MathContainer1, value, NULL);  SetProperty(MathContainer2, value, NULL); end proc:

 

>	calc_ab := proc()  local n, l, a, b;  l := 1;  assume(n, integer);  a[n] := 1/l*int(f(x)*cos(n*Pi*x/l), x=-l..l);  b[n] := 1/l*int(f(x)*sin(n*Pi*x/l), x=-l..l);  SetProperty(MathContainer0, value, a[n]);  SetProperty(MathContainer1, value, b[n]); end proc:

 

>

cb := proc()  global c, f1, f2, f;  if (evalb(GetProperty(ComboBox0, value) = "Step")) then    c := 0: f1 := 0: f2 := 1:  elif(evalb(GetProperty(ComboBox0, value) = "Triangle")) then    c := 0: f1 := x+1: f2 := -x+1:  elif(evalb(GetProperty(ComboBox0, value) = "Saw")) then    c := 0: f1 := x+1: f2 := x:  elif(evalb(GetProperty(ComboBox0, value) = "Other1")) then    c := 0: f1 := 0: f2 := -4*(x-1/2)^2+1:  elif(evalb(GetProperty(ComboBox0, value) = "Other2")) then    c := -1/2: f1 := 0: f2 :=(x+1/2)^2:  elif(evalb(GetProperty(ComboBox0, value) = "Other3")) then    c := 1: f1 := x^2: f2 :=x^2:  end if;  f := x -> piecewise(-1 <= x and x <= c, f1, c <= x and x <= 1, f2);  SetProperty(TextArea0, value, c);  SetProperty(TextArea1, value, c);  SetProperty(TextArea2, value, f1);  SetProperty(TextArea3, value, f2);  SetProperty(Plot0, value, plot(f(x), x=-1..1, thickness=2, scaling=constrained)); end proc:

 

>

calc_fourier := proc()  local N, l, n, a, b;  global g;  calc_ab();  l := 1;  assume(n, integer);  a[n] := 1/l*int(f(x)*cos(n*Pi*x/l), x=-l..l);  b[n] := 1/l*int(f(x)*sin(n*Pi*x/l), x=-l..l);  a[0] := 1/l*int(f(x), x=-l..l);  N := parse(GetProperty(TextArea4, value));  g := (x, N) ->  a[0]/2+sum(a[n]*cos(n*Pi*x/l)+b[n]*sin(n*Pi*x/l), n=1..N);  SetProperty(MathContainer2, value, g(x, N)); end proc:

 

>	draw_fourier := proc()  local N;  N := parse(GetProperty(TextArea4, value));  SetProperty(Plot0, value, fg_plot(N)); end proc:

 

>	fg_plot := proc(N::integer)  plot([f(x), g(x, N)], x=-1..1, thickness=2, scaling=constrained); end proc:

 

>	anim := proc()  calc_fourier();  SetProperty(Plot0, value, plots[display](seq(fg_plot(i), i=1..parse(GetProperty(TextArea4, value))), insequence=true)); end proc:

 

>

 

(1.1)

 

>

 

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