Fourier Transform

$f(x)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }\tilde{f}(k)e^{ikx}\mathrm{d}k$

• The Fourier transform of a reciprocal Space function

$\mathcal{F}[f(x)]=\tilde{f}(k)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }f(x)e^{-ikx}\mathrm{d}x$

• This is the Fourier transform of $f(x)$ that takes inputs from $k$ space

${\int }_{-\infty }^{\infty }f(t)\mathrm{d}t=\tilde{f}(0)$

• 3b1b video

$\mathcal{F}[f(t)\cdot g(t)]=\mathcal{F}[f(t)]\ast \mathcal{F}[g(t)]$

Quantum Physics

Topics

• Momentum Basis
• Borwein Integrals

$\Psi (x)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }\tilde{\Psi }(k)e^{ikx}\mathrm{d}k$

$\tilde{\Psi }(k)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }\Psi (x)e^{-ikx}\mathrm{d}x$

Signal Processing

$F(j\omega )={\int }_{-\infty }^{\infty }f(t)e^{-j\omega t}\mathrm{d}t$

• The $F(j\omega )$ is effectively the same as $F(\omega )$

$f(t)=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }F(j\omega )e^{j\omega t}\mathrm{d}\omega$

Topics

• Fourier Transform Properties
• Fourier Transform Symmetries
• Fourier Transform Catalog