 
 
 
5.25.1  Fourier coefficients : fourier_an and fourier_bn or fourier_cn
 
Let f be a T-periodic continuous functions on
ℝ except maybe at a finite number of points.
One can prove that if f is continuous at x, then;
where the coefficients an, bn, n∈ N, (or cn, n ∈ Z) are the 
Fourier coefficients of f.
The commandsfourier_an and fourier_bn or fourier_cn 
compute these coefficients. 
fourier_an
fourier_an takes four or five arguments : an expression expr
depending on a variable, the name of this variable (for example x), the 
period T, an integer n and a real a (by default a=0).
fourier_an(expr,x,T,n,a) returns the Fourier coefficient an of a 
function f of 
variable x defined on [a,a+T) by f(x)=expr and such that
f is periodic of period T:
To simplify the computations, one should input assume(n,integer) 
before calling fourier_an to specify 
that n is an integer.
Example Let the function f, of period T=2, defined on [−1,1) by 
f(x)=x2.
Input, to have the coefficient a0 :
fourier_an(x^2,x,2,0,-1)
Output :
1/3
Input, to have the coefficient an (n≠ 0) :
assume(n,integer);fourier_an(x^2,x,2,n,-1)
Output :
4*(-1)^n/(pi^2*n^2)
fourier_bn
fourier_bn takes four or five arguments : an expression expr
depending on a variable, the name of this variable (for example x), the 
period T, an integer n and a real a (by default a=0).
fourier_bn(expr,x,T,n,a) returns the Fourier coefficient bn of a 
function f of variable x defined on [a,a+T) by f(x)=expr and periodic 
of period T:
To simplify the computations, one should input assume(n,integer) 
before calling fourier_bn to specify that n is an integer.
Examples 
- 
Let the function f, of period T=2, defined on [−1,1) by 
f(x)=x2.
 Input, to have the coefficient bn (n≠ 0) :assume(n,integer);fourier_bn(x^2,x,2,n,-1)
 Output :0 
- Let the function f, of period T=2, defined on [−1,1) by
f(x)=x3.
 Input, to have the coefficient b1 :fourier_bn(x^3,x,2,1,-1)
 Output :(2*pi^2-12)/pi^3
 
fourier_cn
fourier_cn takes four or five arguments : an expression expr
depending of a variable, the name of this variable (for example x), the 
period T, an integer n and a real a (by default a=0).
fourier_cn(expr,x,T,n,a) returns the Fourier coefficient cn of a 
functionf of variable x defined on [a,a+T) by f(x)=expr and periodic 
of period T:
To simplify the computations, one should
input assume(n,integer) before calling fourier_cn
to specify that n is an integer.
Examples
- 
Find the Fourier coefficients cn of the periodic function f of
period 2 and defined on [−1,1) by  f(x)=x2.
 Input, to have c0 :fourier_cn(x^2,x,2,0,-1)
 Output:1/3 Input, to have cn :assume(n,integer) fourier_cn(x^2,x,2,n,-1)
 Output:2*(-1)^n/(pi^2*n^2)
 
- Find the Fourier coefficients cn of the periodic function f, of 
period 2, and defined on [0,2) by  f(x)=x2.
 Input, to have c0 :fourier_cn(x^2,x,2,0)
 Output:4/3 Input, to have cn :assume(n,integer) fourier_cn(x^2,x,2,n)
 Output:((2*i)*pi*n+2)/(pi^2*n^2)
 
- Find the Fourier coefficients cn of the periodic function f of 
period 2π and defined on [0,2π) by  f(x)=x2.
 Input  :assume(n,integer) fourier_cn(x^2,x,2*pi,n)
 Output :((2*i)*pi*n+2)/n^2
 If you don’t specify assume(n,integer), the output will not be 
simplified :((2*i)*pi^2*n^2*exp((-i)*n*2*pi)+2*pi*n*exp((-i)*n*2*pi)+
 (-i)*exp((-i)*n*2*pi)+i)/(pi*n^3)
 You might simplify this expression by replacing 
exp((-i)*n*2*pi) by 1, input :subst(ans(),exp((-i)*n*2*pi)=1) Output :((2*i)*pi^2*n^2+2*pi*n+-i+i)/pi/n^3
 This expression is then simplified with normal, the final
output is :((2*i)*pi*n+2)/n^2
 Hence for n ≠ 0,  cn=2inπ+2/n2.
As shown in this example, it is better to input assume(n,integer) before calling fourier_cn.
 We must also compute cn for n=0, input :fourier_cn(x^2,x,2*pi,0)
 Output :4*pi^2/3
 Hence for n= 0,  c0=4π2/3.
Remarks :
- 
Input purge(n) to remove the hypothesis done
on n.
- Input about(n) or assume(n), to know 
the hypothesis done on the variable n.
 
 
