Fourier coefficients

### 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;

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 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 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:

  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 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:

  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 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:

  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:

fourier_an(x^2,x,2,0,-1)

1
3

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

assume(n,integer):; fourier_an(x^2,x,2,n,-1)

4 | ⎛ ⎝ | −1 | ⎞ ⎠ | n
n2 π 2

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

assume(n,integer):; fourier_bn(x^2,x,2,n,-1)

0 |  |  |  |  |  |  |  |  |  |

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

fourier_bn(x^3,x,2,1,-1)

2 π 2−12
π 3

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

fourier_cn(x^2,x,2,0,-1)

1
3

Input (to get cn):

assume(n,integer); fourier_cn(x^2,x,2,n,-1)

2 | ⎛ ⎝ | −1 | ⎞ ⎠ | n
n2 π 2

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

fourier_cn(x^2,x,2,0)

4
3

To get cn, input:

assume(n,integer):; fourier_cn(x^2,x,2,n)

2 π i n+2
n2 π 2

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

assume(n,integer):; fourier_cn(x^2,x,2*pi,n)

2πi n+2
n2

You must also compute cn for n=0:

fourier_cn(x^2,x,2*pi,0)

4
3
| π 2

##### 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.