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La serie Fourier
(Matemática | Avanzado | Serie Fourier)

La serie de fourier de la función f(x)

a(0) / 2 + (sum)(k=1..inf) (a(k) cos kx + b(k) sin kx)

a(k) = 1/PI (integral)(-PI to PI) f(x) cos kx dx
b(k) = 1/PI (integral)(-PI to PI) f(x) sin kx dx
El residuo de la serie de fourier. Sn(x) = la suma de los primeros n+1 términos a x.

el residuo(n) = f(x) - Sn(x) = 1/PI (integral)(-PI to PI) f(x+t) Dn(t) dt

Sn(x) = 1/PI (integral)(-PI to PI) f(x+t) Dn(t) dt

Dn(x) = Dirichlet kernel = 1/2 + cos x + cos 2x + .. + cos nx = [ sin(n + 1/2)x ] / [ 2sin(x/2) ]
Teorema de Riemann. Si f(x) es continuo a excepción de un número finito de saltos finitos en todos los intervalos finitos pues:

lim(k->inf) (integral)(a-b) f(t) cos kt dt = lim(k->inf)(integral)(a-b)f(t) sin kt dt = 0

La serie fourier de la función f(x) en un intervalo arbitrario.

A(0) / 2 + (sum)(k=1..inf) [ A(k) cos (k(PI)x / m) + B(k) (sin k(PI)x / m) ]

a(k) = 1/m (integral)(-m-&gtm)f(x) cos (k(PI)x / m) dx
b(k) = 1/m (integral)(-m-&gtm)f(x) sin (k(PI)x / m) dx
El Teorema de Parseval. Si f(x) es continuo; f(-PI) = f(PI) pues

1/PI (integral)(-PI to PI) f^2(x) dx = a(0)^2 / 2 + (sum)(k=1..inf) (a(k)^2 + b(k)^2)

La Integral Fourier de la función f(x)

f(x) = (integral)(0-inf) ( a(y) cos yx + b(y) sin yx ) dy

a(y) = 1/PI (integral)(-inf-&gtinf) f(t) cos ty dt
b(y) = 1/PI (integral)(-inf-&gtinf) f(t) sin ty dt
f(x) = 1/PI (integral)(0-inf) dy (integral)(-inf-&thinf)f(t) cos (y(x-t)) dt

Casos espaciales de la Integral Fourier

si f(x) = f(-x) pues

f(x) = 2/PI (integral)(0-inf) cos xy dy (integral)(0-inf)f(t) cos yt dt
if f(-x) = -f(x) then
f(x) = 2/PI (integral)(0-inf) sin xy dy (integral)(0-inf)sin yt dt
(Transforms) de Fourier

(Transform) Fourier coseno

g(x) = sqrt(2/PI)(integral)(0-inf)f(t) cos xt dt

(Transform) Fourier seno

g(x) = sqrt(2/PI)(integral)(0-inf)f(t) sin xt dt

Identidades de los (tranforms)

Si f(-x) = f(x) pues

Transform Fourier coseno ( Tranform Fourier coseno (f(x)) ) = f(x)
Si f(-x) = -f(x) pues
Transform Fourier seno (Transform Fourier seno (f(x)) ) = f(x)


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