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Hilbert transform
Partial Function
| $H: (\mathbb C\to\mathbb C)\to(\mathbb C\to\mathbb C)$ |
| $H(f):=t\mapsto \frac{1}{\pi}\cdot\mathcal P\int_{-\infty}^\infty\frac{f(\tau)}{t-\tau}\,\mathrm dx$ |
Discussion
$H(H(f))=-f$
The Hilbert transform is used in the Kramers–Kronig relation/Sokhotski–Plemelj theorem to express the imaginary part of an analytic function in terms of its real part (or the other way around). This works because they get “mixed up” in the Cauchy integral formula which introduces a factor of $\frac{1}{i}$. The principal value is taken to push the complex line integral on the real line.
Reference
Wikipedia: Hilbert transform
Parents
Requirements
