5.23.1  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 a_n, b_n, n∈ N, (or c_n, 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 a_n 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 :

Output :

Input, to have the coefficient a_n (n≠ 0) :

Output :

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 b_n 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 b_n (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 c_n 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 c_n 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 c_n :
  assume(n,integer)
  fourier_cn(x^2,x,2,n,-1)
  Output:
  2*(-1)^n/(pi^2*n^2)
• Find the Fourier coefficients c_n 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 c_n :
  assume(n,integer)
  fourier_cn(x^2,x,2,n)
  Output:
  ((2*i)*pi*n+2)/(pi^2*n^2)
• Find the Fourier coefficients c_n 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, c_n=2inπ+2/n². As shown in this example, it is better to input assume(n,integer) before calling fourier_cn.
  We must also compute c_n 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.