The constant π also appears as a critical spectral parameter in the Fourier transform. This is the integral transform, that takes a complex-valued integrable function f on the real line to the function defined as:

${\displaystyle \stackrel{^<!--\; ^\; -->}{f}(\mathrm{\xi <!--\; \xi \; -->})={\int <!--\; \int \; -->}_{-<!--\; -\; -->\mathrm{\infty <!--\; \infty \; -->}}^{\mathrm{\infty <!--\; \infty \; -->}}f(x)e^{-<!--\; -\; -->2\mathrm{\pi <!--\; \pi \; -->}ix\mathrm{\xi <!--\; \xi \; -->}}}$

The constant π also appears as a critical spectral parameter in the Fourier transform. This is the integral transform, that takes a complex-valued integrable function f on the real line to the function defined as:

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 L² that is also an algebra homomorphism of L¹ 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,

${\displaystyle ({\int <!--\; \int \; -->}_{-<!--\; -\; -->\mathrm{\infty <!--\; \infty \; -->}}^{\mathrm{\infty <!--\; \infty \; -->}}{x}^2|f(x){|}^{2}dx}$