including Fourier transform

$\mathcal{F}[f(x)]=\frac{1}{\sqrt{2\pi}}{\int^\infty_{-\infty} f(x) e^{-i \xi x} dx$
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$\mathcal{F}[e^{\frac{-x^2}{a}}]=\sqrt{\frac{a}{2e}}e^{-\frac{-a\xi^2}{4}}~~(a>0)$
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$\mathcal{F}[e^{-a|x|}]=\sqrt{\frac{2}{\pi}}{\frac{a}{a^2+\xi^2}}~~(a>0)$
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$\mathcal{F}[\frac{1}{1+x^2}]=\sqrt{\frac{{\pi}}{2}} e^{-|\xi|}$
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$\mathcal{F}[\frac{1}{1+x^4}]=\sqrt{\frac{{\pi}}{2}} e^{-\frac{|\xi|}{\sqrt{2}}} \sin (\frac{|\xi|}{\sqrt{2}}+\frac{\pi}{4})$
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$\mathcal{F}[\frac{e^{-a|x|}}{\sqrt{|x|}}]=\frac{\sqrt{a+\sqrt{a^2+\xi^2}}}{\sqrt{a^2+\xi^2}}~~(a>0)$
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$\mathcal{F}[\log \frac{x^2+a}{x^2+b}]=-\frac{\sqrt{2\pi}}{|\xi|}(e^{-a|\xi|}-e^{-b|\xi|})~~(a\geq0~~b\geq0)$
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$\mathcal{F}[\arctan(\frac{x}{a})]=-\sqrt{\frac{\pi}{2}}i~\mbox{sgn}~a \cdot(\frac{e^{-|a||\xi|}}{\xi})$
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