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Hilbert kernel

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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} $.

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References

[a1] B.L. Moiseiwitsch, "Integral equations" , Longman (1977)
How to Cite This Entry:
Hilbert kernel. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hilbert_kernel&oldid=47231
This article was adapted from an original article by B.V. Khvedelidze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article