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 b_n of a function f of variable x defined on [a, a + T[ by f (x) = expr and periodic of period T:

b_n = $${\frac{{2}}{{T}}}$$$$\int_{a}^{{a+T}}$$f (x)sin($${\frac{{2\pi nx}}{{T}}}$$)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 b_n (n $\neq$ 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