6.5 Fourier transformation of distributions
δ(x) dx = 1.
Hence we have
dk = δ(x).
Of course, it is extremely hard to make sense of this integral. It quite obviously
diverges as a normal integral, but we can just have faith and believe this makes
sense as long as we are talking about distributions.
Similarly, from our rules of translations, we get
F[δ(x − a)] = e
] = 2πδ(k − `),
Hence we get
F[cos(`x)] = F
] = π[δ(k−`)+δ(k+`)].
We see that highly localized functions in
-space have very spread-out behaviour
in k-space, and vice versa.