Matematik
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Fourier Dönüşüm Formülleri
- \(f(x)=\int^{\infty}_{-\infty}{F(k)e^{2\pi ikx}dk}\quad;\quad F(k)=\int^{\infty}_{-\infty}{f(x)e^{-2\pi ikx}dx}\)
- \(F(k)=\mathcal{F}_x[f(x)](k)=\int^{\infty}_{-\infty}{f(x)e^{-2\pi ikx}dx}\)
- \(f(x)=\mathcal{F}^{-1}_k[F(k)](x)=\int^{\infty}_{-\infty}{F(k)e^{2\pi ikx}dk}\)
- \(\mathcal{F}[f(x)]=F(k)\quad;\quad \mathcal{F}[g(x)]=G(k)\;\;\text{ise}\)
- 1.
- \(\mathcal{F}[af(x)+bg(x)]=a\mathcal{F}[f(x)]+b\mathcal{F}[g(x)]=aF(k)+bF(k)\)
- 2.
- \(\mathcal{F}_x[1](k)=\int^{\infty}_{-\infty}{f(x)e^{-2\pi ikx}dx}=\delta(k)\)
- 3.
- \(\begin{align*} \mathcal{F}[\sin(2\pi k_0 x)](k) &= \frac{1}{2}i[\delta(k+k_0)-\delta(k-k_0)] \end{align*}\)
- 4.
- \(\begin{align*} \mathcal{F}[\delta(x-x_0)](k) &= \int^{\infty}_{-\infty}{\delta(x-x_0) e^{-2\pi ikx}dx}\\ &= e^{-2\pi ikx_0} \end{align*}\)
- 5.
- \(\begin{align*} \mathcal{F}[e^{-k_0|x|}](k) &= \int^{\infty}_{-\infty}{e^{-k_0|x|} e^{-2\pi ikx}dx}\\ &= \frac{1}{\pi}\frac{k_0}{k^2+k^2_0} \end{align*}\)
- 6.
- \(\begin{align*} \mathcal{F}[e^{-ax^2}](k) &= \int^{\infty}_{-\infty}{e^{-ax^2} e^{-2\pi ikx}dx}\\ &= \sqrt{\frac{\pi}{a}}e^{-\pi^2k^2/a} \end{align*}\)
- 7.
- \(\begin{align*} \mathcal{F}[H(x)](k) &= \int^{\infty}_{-\infty}{e^{-2\pi ikx}H(x)dx}\\ &= \frac{1}{2}\left[\delta(k)-\frac{i}{\pi k} \right ] \end{align*}\)
- 8.
- \(\begin{align*} \mathcal{F}[f'(x)](k) &= 2\pi ik\mathcal{F}[f(x)](k) \end{align*}\)
- 9.
- \(\begin{align*} \mathcal{F}[f^{(n)}(x)](k) &= (2\pi ik)^n\mathcal{F}[f(x)](k) \end{align*}\)
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