Hilbert kernel

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)