Difference between revisions of "Hilbert kernel"
From Encyclopedia of Mathematics
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| − | The kernel of the [[ | + | The kernel of the [[Hilbert singular integral]], i.e. the function |
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| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> B.L. Moiseiwitsch, "Integral equations" , Longman (1977)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> B.L. Moiseiwitsch, "Integral equations" , Longman (1977)</TD></TR> | ||
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Latest revision as of 20:27, 18 March 2024
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) |
How to Cite This Entry:
Hilbert kernel. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hilbert_kernel&oldid=55653
Hilbert kernel. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hilbert_kernel&oldid=55653
This article was adapted from an original article by B.V. Khvedelidze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article