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Difference between revisions of "Cauchy kernel"

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The term refers usual 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|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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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  
 
\[
 
\[
 
\frac{d\zeta}{\zeta-z} = \frac{1}{2} \left(\cot \frac{\tau-t}{2} + i\right)\, d\tau\, .
 
\frac{d\zeta}{\zeta-z} = \frac{1}{2} \left(\cot \frac{\tau-t}{2} + i\right)\, d\tau\, .

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=31229
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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