21.4.1  Fourier coefficients

Let f be a T-periodic continuous function on ℝ except perhaps 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 fourier_an and fourier_bn or fourier_cn commands compute these coefficients.

• fourier_an takes four mandatory and one optional argument:
  • expr, an expression depending on a variable.
  • x, the name of this variable.
  • T, the period.
  • n, a non-negative integer.
  • Optionally, a, a real number (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 with period T:

  an=

  2 T

  ∫

  a+T a

  f(x)cos

  ⎛ ⎜ ⎜ ⎝

  2π nx T

  ⎞ ⎟ ⎟ ⎠

  dx

• fourier_bn takes four mandatory and one optional argument:
  • expr, and expression depending on a variable.
  • x, the name of this variable.
  • T, the period.
  • n, an integer.
  • Optionally, a, a real number (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 such that f is periodic with period T:

  bn=

  2 T

  ∫

  a+T a

  f(x)sin

  ⎛ ⎜ ⎜ ⎝

  2π nx T

  ⎞ ⎟ ⎟ ⎠

  dx

• fourier_cn takes four mandatory and one optional argument:
  • expr, and expression depending on a variable.
  • x, the name of this variable.
  • T, the period.
  • n, an integer.
  • Optionally, a, a real number (by default a=0).
• fourier_cn(expr,x,T,n ⟨,a⟩) returns the Fourier coefficient c_n of a function f of variable x defined on [a,a+T) by f(x)=expr and such that f is periodic with period T:

  cn=

  1 T

  ∫

  a+T a

  f(x) e

  −2iπ nx T

  dx

To simplify the computations, you should input assume(n,integer) (see Section 3.3.8) before calling the above commands with an unspecified n to specify that it is an integer.

Examples

Let the function f, with period T=2, be defined on [−1,1) by f(x)=x². To obtain the coefficient a₀, input:

To obtain the coefficient a_n (n≠ 0), input:

Let the function f, with period T=2, defined on [−1,1) by f(x)=x². To get the coefficient b_n (n≠ 0), input:

Let the function f, with period T=2, defined on [−1,1) by f(x)=x³. To get the coefficient b₁, input:

Find the Fourier coefficients c_n of the periodic function f of period 2 and defined on [−1,1) by f(x)=x². To get c₀, input:

Input (to get c_n):

Find the Fourier coefficients c_n of the periodic function f, of period 2, and defined on [0,2) by f(x)=x². To get c₀, input:

To get c_n, input:

Find the Fourier coefficients c_n of the periodic function f of period 2π and defined on [0,2π) by f(x)=x².

You must also compute c_n for n=0:

Remarks.

• Input purge(n) (see Section 3.3.9) to remove the hypothesis done on n.
• Input about(n) or assume(n), to know the hypothesis done on the variable n.