Fourier transformation
Symmetric Fourier transformation $$ \mathcal{F}f = \frac{1}{\sqrt{2\pi}^n}\int_{\Reals^n}f(x)e^{-iyx}dx$$ inverse $$f(x) = \frac{1}{\sqrt{2\pi}^n}\int_{\Reals^n} \mathcal{F}fe^{iyx}dy$$ asymmetric Fourier transformation $$ \mathcal{F}f = \int_{\Reals^n}f(x)e^{-iyx}dx$$ inverse $$f(x) = \frac{1}{(2\pi)^n}\int_{\Reals^n} \mathcal{F}fe^{iyx}dy$$
In the physical application case we usually transform between the position $\vec{x}$ and momentum space $\vec{p}$ so that $$ \mathcal{F}f = \int f(x)e^{-i\vec{p}\vec{x}}d^3p$$ inverse $$f(\vec{x}) = \frac{1}{(2\pi)^n}\int \mathcal{F}fe^{i\vec{p}\vec{x}}d^3p$$
Differentiation
From partial integration one can derive the transformation of the $n$th derivative $$ \mathcal{F} D^n f = i^{|n|}|x|^n \mathcal{F}f$$