Fourier coefficients : fourier_an and fourier_bn or fourier_cn

### 2.22.1 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;

f(x) | = |
a0
2
| + |
+∞
∑
n=1
| an cos( |
2π nx
T
| )+bn sin( |
2π nx
T
| )
| = |
+∞
∑
n=−∞
| cn e |
2iπ nx
T

where the coefficients an, bn, n∈ N, (or cn, n ∈ Z) are the Fourier coefficients of f. The commands`fourier_an` and `fourier_bn` or `fourier_cn` compute these coefficients.

#### `fourier_an`

`fourier_an` 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_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:

an= |
2
T
| ∫ |
a+T
a
| f(x)cos( |
2π nx
T
| )dx

To simplify the computations, ons 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 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_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:

bn= |
2
T
| ∫ |
a+T
a
| f(x)sin( |
2π nx
T
| )dx

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:

cn= |
1
T
| ∫ |
a+T
a
| f(x)e |
−2iπ nx
T
| dx

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 2.π and defined on [0;2.π[ 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=4.π2/3.

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.