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Ukrainian Mathematical Journal

, Volume 60, Issue 11, pp 1683–1690 | Cite as

A property of the β-Cauchy-type integral with continuous density

  • R. Abreu Blaya
  • J. Bory ReyesEmail author
Article

A theorem from the classical complex analysis proved by Davydov in 1949 is extended to the theory of solution of a special case of the Beltrami equation in the z-complex plane (i.e., null solutions of the differential operator \(\partial _{{\overline{z} }} - {\upbeta} \frac{Z}{{\overline{Z} }}\partial _{z} ,\;0 \leq {\upbeta} < 1\)).

It is proved that if γ is a rectifiable Jordan closed curve and f is a continuous complex-valued function on γ such that the integral
$${\int\limits_{{\upgamma}\backslash {\left\{ {{\upzeta} \in {\upgamma} :{\left| {{\upzeta} - t} \right|} \leq r} \right\}}} {\frac{{{\left| {f{\left( {\upzeta} \right)} - f{\left( t \right)}} \right|}}}{{{\left| {{\upzeta} - t{\left| {t \mathord{\left/ {\vphantom {t {\upzeta}}} \right. \kern-\nulldelimiterspace} {\upzeta}} \right|}^{{\uptheta}} } \right|}}}{\left| {{\left( {n{\left( {\upzeta} \right)} - {\upbeta} \frac{{\upzeta}}{{\overline{{\upzeta}} }}\overline{n} {\left( {\upzeta} \right)}} \right)}} \right|}ds,\quad {\uptheta} = \frac{{2{\upbeta}}}{{1 - {\upbeta}}},} }$$
converges uniformly on γ as r → 0, where n(ζ) is the unit vector of outer normal on γ at a point ζ and ds is the differential of arc length, then the β-Cauchy-type integral
$$\frac{1}{{2{\left( {1 - {\upbeta} } \right)}{\uppi} }}{\int\limits_{\mathrm \gamma} {\frac{{f{\left( {\upzeta} \right)}}}{{{\upzeta} - z{\left| {z \mathord{\left/ {\vphantom {z {\upzeta} }} \right. \kern-\nulldelimiterspace} {\upzeta} } \right|}^{\uptheta } }}{\left( {n{\left( {\upzeta} \right)} - {\upbeta} \frac{{\upzeta} }{{\overline{{\upzeta} } }}\overline{n} {\left( {\upzeta} \right)}} \right)}ds,\quad z \notin {\upgamma} ,} }$$
admits a continuous extension to γ and a version of the Sokhotski–Plemelj formulas holds.

Keywords

Continuous Limit Continuous Density Beltrami Equation Regular Curve Lipschitz Class 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.Universidad de HolguínHolguínCuba
  2. 2.Universidad de OrienteSantiago de CubaCuba

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