Hilbert kernel
From HandWiki
The kernel of the Hilbert singular integral, i.e. the function
$$
\mathop{\rm cotan} {
\frac{x - s }{2}
} ,\ \
0 \leq x, s \leq 2 \pi . $$
The following simple relation holds between the Hilbert kernel and the Cauchy kernel in the case of the unit circle:
$$
\frac{dt }{t - \tau }
= {
\frac{1}{2}
}
\left ( \mathop{\rm cotan} { \frac{x - s }{2}
} + i \right ) dx,
$$
where $ t = e ^ {ix} $, $ \tau = e ^ {is} $.
References
| [a1] | B.L. Moiseiwitsch, "Integral equations" , Longman (1977) |
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