6.26.1 Fourier coefficients: fourier_an and fourier_bn or fourier_cn
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 an, bn, n∈ N, (or cn, n ∈ Z) are the
Fourier coefficients of f.
The fourier_an and fourier_bn or
fourier_cn commands compute these coefficients.
fourier_an
-
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 an 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:
To simplify the computations, you should input assume(n,integer)
(see Section 5.4.8) before calling fourier_an with an
unspecified n to specify that it is an integer.
Example.
Let the function f, with period T=2, be defined on [−1,1) by f(x)=x2.
Input (to have the coefficient a0):
fourier_an(x^2,x,2,0,-1)
Output:
Input (to have the coefficient an (n≠ 0)):
assume(n,integer) |
fourier_an(x^2,x,2,n,-1)
|
Output:
fourier_bn
-
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 bn 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:
To simplify the computations, you should input assume(n,integer)
(see Section 5.4.8) before calling fourier_bn to
specify that n is an integer.
Examples.
-
Let the function f, with period T=2, defined on [−1,1) by
f(x)=x2.
Input (to get the coefficient bn (n≠ 0)):
assume(n,integer) |
fourier_bn(x^2,x,2,n,-1)
|
Output:
- Let the function f, with period T=2, defined on [−1,1) by
f(x)=x3.
Input (to get the coefficient b1):
fourier_bn(x^3,x,2,1,-1)
Output:
fourier_cn
-
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 cn 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:
To simplify the computations, you should input assume(n,integer)
(see Section 5.4.8) 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 get c0):
fourier_cn(x^2,x,2,0,-1)
Output:
Input (to get cn):
assume(n,integer) |
fourier_cn(x^2,x,2,n,-1)
|
Output:
- 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:
Input (to get cn):
assume(n,integer) |
fourier_cn(x^2,x,2,n)
|
Output:
- 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:
You must also compute cn for n=0:
Input:
fourier_cn(x^2,x,2*pi,0)
Output:
Hence for n= 0, c0=4π2/3.
Remarks.
-
Input purge(n) (see Section 5.4.9) to
remove the hypothesis done on n.
- Input about(n) or assume(n), to know
the hypothesis done on the variable n.