Matematica
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Trasformazione di Fourier
- f(x)=∫∞−∞F(k)e2πikxdk;F(k)=∫∞−∞f(x)e−2πikxdx
- F(k)=Fx[f(x)](k)=∫∞−∞f(x)e−2πikxdx
- f(x)=F−1k[F(k)](x)=∫∞−∞F(k)e2πikxdk
- F[f(x)]=F(k);F[g(x)]=G(k)ise
- 1.
- F[af(x)+bg(x)]=aF[f(x)]+bF[g(x)]=aF(k)+bF(k)
- 2.
- Fx[1](k)=∫∞−∞f(x)e−2πikxdx=δ(k)
- 3.
- F[sin(2πk0x)](k)=12i[δ(k+k0)−δ(k−k0)]
- 4.
- F[δ(x−x0)](k)=∫∞−∞δ(x−x0)e−2πikxdx=e−2πikx0
- 5.
- F[e−k0|x|](k)=∫∞−∞e−k0|x|e−2πikxdx=1πk0k2+k20
- 6.
- F[e−ax2](k)=∫∞−∞e−ax2e−2πikxdx=√πae−π2k2/a
- 7.
- F[H(x)](k)=∫∞−∞e−2πikxH(x)dx=12[δ(k)−iπk]
- 8.
- F[f′(x)](k)=2πikF[f(x)](k)
- 9.
- F[f(n)(x)](k)=(2πik)nF[f(x)](k)
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