$\hat{f}(\xi )={\int}_{-\mathrm{\infty}}^{\mathrm{\infty}}f(x){e}^{-2\pi i<}$Although there are several different conventions for the Fourier transform and its inverse, any such convention must involve π *somewhere*. The above is the most canonical definition, however, giving the unique unitary operator on *L*^{2} that is also an algebra homomorphism of *L*^{1} to *L*^{∞}.^{[164]}

The Heisenberg uncertainty principle also contains the number π. The uncertainty principle gives a sharp lower bound on the extent to which it is possible to localize a function both in space and in frequency: with our conventions for the Fourier transform,

- $({\int}_{-\mathrm{\infty}}^{\mathrm{\infty}}{x}^{2}|f(x){|}^{2}\phantom{\rule{thinmathspace}{0ex}}dx$