<P> There are several different conventions for the Fourier transform, all of which involve a factor of π that is placed somewhere . The appearance of π is essential in these formulas, as there is there is no possibility to remove π altogether from the Fourier transform and its inverse transform . The definition given above is the most canonical, however, because it describes the unique unitary operator on L that is also an algebra homomorphism of L to L . </P> <P> 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, </P> <Dl> <Dd> ∫ − ∞ ∞ x 2 f (x) 2 d x ∫ − ∞ ∞ ξ 2 f ^ (ξ) 2 d ξ ≥ (1 4 π ∫ − ∞ ∞ f (x) 2 d x) 2 . (\ displaystyle \ int _ (- \ infty) ^ (\ infty) x ^ (2) f (x) ^ (2) \, dx \ \ int _ (- \ infty) ^ (\ infty) \ xi ^ (2) (\ hat (f)) (\ xi) ^ (2) \, d \ xi \ geq \ left ((\ frac (1) (4 \ pi)) \ int _ (- \ infty) ^ (\ infty) f (x) ^ (2) \, dx \ right) ^ (2).) </Dd> </Dl> <Dd> ∫ − ∞ ∞ x 2 f (x) 2 d x ∫ − ∞ ∞ ξ 2 f ^ (ξ) 2 d ξ ≥ (1 4 π ∫ − ∞ ∞ f (x) 2 d x) 2 . (\ displaystyle \ int _ (- \ infty) ^ (\ infty) x ^ (2) f (x) ^ (2) \, dx \ \ int _ (- \ infty) ^ (\ infty) \ xi ^ (2) (\ hat (f)) (\ xi) ^ (2) \, d \ xi \ geq \ left ((\ frac (1) (4 \ pi)) \ int _ (- \ infty) ^ (\ infty) f (x) ^ (2) \, dx \ right) ^ (2).) </Dd>

Pi is the ratio of circumference to diameter