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The kernel of the [[Hilbert singular integral|Hilbert singular integral]], i.e. the function
 
The kernel of the [[Hilbert singular integral|Hilbert singular integral]], i.e. the function
  
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047300/h0473001.png" /></td> </tr></table>
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$$
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\mathop{\rm cotan}  {
 +
\frac{x - s }{2}
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} ,\ \
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0 \leq  x, s \leq  2 \pi .
 +
$$
  
 
The following simple relation holds between the Hilbert kernel and the [[Cauchy kernel|Cauchy kernel]] in the case of the unit circle:
 
The following simple relation holds between the Hilbert kernel and the [[Cauchy kernel|Cauchy kernel]] in the case of the unit circle:
  
−
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047300/h0473002.png" /></td> </tr></table>
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$$
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047300/h0473003.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047300/h0473004.png" />.
 
  
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\frac{dt }{t - \tau }
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  =  {
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\frac{1}{2}
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}
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\left (  \mathop{\rm cotan}  {
 +
\frac{x - s }{2}
 +
} + i \right )  dx,
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$$
  
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where  $  t = e  ^ {ix} $,
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$  \tau = e  ^ {is} $.
  
 
====Comments====
 
====Comments====
−
 
  
 
====References====
 
====References====
 
<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></table>

Revision as of 22:10, 5 June 2020


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} $.

Comments

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
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