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

Stein 

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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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