These are pre-computed Fourier Transform pairs for commonly used functions. They act as reference identities to simplify transformations without computing integrals manually.
\[ f(x) \iff F(s) = \int_{-\infty}^{\infty} f(x)e^{-2\pi i sx}dx \]
\[ f(ax) \iff \frac{1}{|a|}F\left(\frac{s}{a}\right) \]
\[ f(x-a) \iff e^{-2\pi ias}F(s) \]
\[ \frac{d^n}{dx^n} f(x) \iff (2\pi is)^n F(s) \]
\[ \delta(x) \iff 1 \quad ; \quad 1 \iff \delta(s) \]
\[ e^{-a|x|} \iff \frac{2a}{a^2 + 4\pi^2 s^2} \quad ; \quad xe^{-a|x|} \iff \frac{8\pi ia s}{(a^2 + 4\pi^2 s^2)^2} \]
\[ e^{-x^2/a^2} \iff a\sqrt{\pi}e^{-\pi^2 a^2 s^2} \]
\[ \sin(ax) \iff \frac{1}{2i}[\delta(s - a/2\pi) - \delta(s + a/2\pi)] \]
\[ \cos(ax) \iff \frac{1}{2}[\delta(s - a/2\pi) + \delta(s + a/2\pi)] \]
\[ \text{sinc}(x) = \frac{\sin(\pi x)}{\pi x} \iff \text{Rectangular pulse} \]