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Difference between revisions of "Cauchy kernel"

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The function of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020910/c0209101.png" />, which is the kernel of the [[Cauchy integral|Cauchy integral]]. In the case of the unit circle one has the following relationship between the Cauchy kernel and the [[Hilbert kernel|Hilbert kernel]]:
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{{TEX|done}}
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{{MSC|30-XX}}
  
<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/c/c020/c020910/c0209102.png" /></td> </tr></table>
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The term refers usually to the function $\mathbb C^2 \setminus \Delta \ni (\zeta, z) \mapsto \frac{1}{\zeta-z}$, where $\Delta \subset \mathbb C^2$ is the diagonal $\{(z,\zeta): z=\zeta\}$. Such function is the [[Kernel function|kernel]] of the [[Cauchy integral]], which gives a powerful identity for [[Holomorphic function|holomorphic functions]] of one complex variable. In the case of the unit circle one has the following relationship between the Cauchy kernel and the [[Hilbert kernel]]: if $\zeta = e^{i\tau}$ and $z = e^{it}$, with $\tau, t \in \mathbb S^1$, then
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\[
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\frac{d\zeta}{\zeta-z} = \frac{1}{2} \left(\cot \frac{\tau-t}{2} + i\right)\, d\tau\, .
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\]
  
where
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Some authors use the term for the function
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\[
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\frac{1}{2\pi i (\zeta-z)}\, .
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\]
  
<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/c/c020/c020910/c0209103.png" /></td> </tr></table>
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See also [[Kernel of an integral operator]].
 
 
The term Cauchy kernel is sometimes applied to the function
 
 
 
<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/c/c020/c020910/c0209104.png" /></td> </tr></table>
 
 
 
 
 
 
 
====Comments====
 
See also [[Kernel function|Kernel function]]; [[Kernel of an integral operator|Kernel of an integral operator]].
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"A.I. Markushevich,   "Theory of functions of a complex variable" , '''1–3''' , Chelsea (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"E.C. Titchmarsh,   "Introduction to the theory of Fourier integrals" , Oxford Univ. Press (1948)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> W. Rudin,  "Real and complex analysis" , McGraw-Hill  (1974)  pp. 24</TD></TR></table>
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{|
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|-
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|valign="top"|{{Ref|Al}}|| L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1966) {{MR|0188405}} {{ZBL|0154.31904}}
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|-
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|valign="top"|{{Ref|Ma}}|| A.I. Markushevich, "Theory of functions of a complex variable" , '''1–3''' , Chelsea (1977)  (Translated from Russian) {{MR|0444912}}  {{ZBL|0357.30002}}
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|-
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|valign="top"|{{Ref|Ru}}|| W. Rudin, "Real and complex analysis" , McGraw-Hill (1974) pp. 24 {{MR|0344043}} {{ZBL|0278.26001}}
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|-
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|valign="top"|{{Ref|Ti}}|| E.C. Titchmarsh, "The theory of functions" , Oxford Univ. Press (1939{{MR|0593142}} {{MR|0197687}}  {{MR|1523319}} {{ZBL|65.0302.01}}
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|-
 +
|}

Latest revision as of 20:26, 18 March 2024

2020 Mathematics Subject Classification: Primary: 30-XX [MSN][ZBL]

The term refers usually to the function $\mathbb C^2 \setminus \Delta \ni (\zeta, z) \mapsto \frac{1}{\zeta-z}$, where $\Delta \subset \mathbb C^2$ is the diagonal $\{(z,\zeta): z=\zeta\}$. Such function is the kernel of the Cauchy integral, which gives a powerful identity for holomorphic functions of one complex variable. In the case of the unit circle one has the following relationship between the Cauchy kernel and the Hilbert kernel: if $\zeta = e^{i\tau}$ and $z = e^{it}$, with $\tau, t \in \mathbb S^1$, then \[ \frac{d\zeta}{\zeta-z} = \frac{1}{2} \left(\cot \frac{\tau-t}{2} + i\right)\, d\tau\, . \]

Some authors use the term for the function \[ \frac{1}{2\pi i (\zeta-z)}\, . \]

See also Kernel of an integral operator.

References

[Al] L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1966) MR0188405 Zbl 0154.31904
[Ma] A.I. Markushevich, "Theory of functions of a complex variable" , 1–3 , Chelsea (1977) (Translated from Russian) MR0444912 Zbl 0357.30002
[Ru] W. Rudin, "Real and complex analysis" , McGraw-Hill (1974) pp. 24 MR0344043 Zbl 0278.26001
[Ti] E.C. Titchmarsh, "The theory of functions" , Oxford Univ. Press (1939) MR0593142 MR0197687 MR1523319 Zbl 65.0302.01
How to Cite This Entry:
Cauchy kernel. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cauchy_kernel&oldid=18064
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article