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Prelude

  • Fausto Di Biase
Part of the Progress in Mathematics book series (PM, volume 147)

Abstract

Let U be an open subset of the complex plane ℂ. A smooth function g: U → ℝ is harmonic if the Laplace equation
$$\frac{{{\partial ^2}g}}{{\partial {x^2}}} + \frac{{{\partial ^2}g}}{{\partial {y^2}}} = 0$$
holds at every point of U. A smooth function Φ: U → ℂ is holomorphic if the real and imaginary parts g and h of Φ satisfy the Cauchy-Riemann equations
$$\frac{{\partial g}}{{\partial x}} = \frac{{\partial h}}{{\partial y}},\frac{{\partial g}}{{\partial y}} = - \frac{{\partial h}}{{\partial x}}$$
in U; then g and h are harmonic, and h is said to be a conjugate harmonic function of g. It can be shown that Φ is holomorphic if and only if it is representable in U by the power series \(\sum _0^\infty {c_n}{\left( {z - {z_0}} \right)^n} \equiv \Phi \left( z \right)\) for each \({z_0} \in U,|z - {z_0}| < {\inf _{\zeta \notin U}}|\zeta - {z_0}|.\)

Keywords

Unit Disc Maximal Function Approach Region Homogeneous Type Poisson Kernel 
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

© Birkhäuser 1998

Authors and Affiliations

  • Fausto Di Biase
    • 1
    • 2
  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Dip. MatematicaUniversity Roma-Tor VergataRomeItaly

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