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

c_n = $${\frac{{1}}{{T}}}$$$$\int_{a}^{{a+T}}$$f (x)e^{${\frac{}{}}$}-2i$$\pi$$nxTdx

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 c_n 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 c_n :
  assume(n,integer)
  fourier_cn(x^2,x,2,n,-1)
  Output:
  2*(-1)^n/(pi^2*n^2)

• Find the Fourier coefficients c_n 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 c_n :
  assume(n,integer)
  fourier_cn(x^2,x,2,n)
  Output:
  ((2*i)*pi*n+2)/(pi^2*n^2)

• Find the Fourier coefficients c_n of the periodic function f of period 2.$\pi$ and defined on [0;2.$\pi$[ 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 $\neq$ 0, c_n = $${\frac{{2in\pi+2}}{{n^2}}}$$. As shown in this example, it is better to input assume(n,integer) before calling fourier_cn.
  We must also compute c_n for n = 0, input :
  fourier_cn(x^2,x,2*pi,0)
  Output :
  4*pi^2/3
  Hence for n = 0, c₀ = $${\frac{{4.{\pi}^2}}{{3}}}$$.

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.