Harmonic functions in three variables

  • Stefan Bergman
Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE2, volume 23)


As indicated in the introduction, the method of integral operators is one of the tools which may be used to extend procedures of the theory of functions of one and several complex variables to other fields, in particular to the theory of harmonic functions in three variables. The totality {U} of these functions represent a linear space, but they neither form an algebra nor possess group properties. However, by associating, in a one-to-one way, a function f k of one or more complex variables with every hormonic function of three variables, \( {U_k} = {\tilde P_3}({f_k}),{\rm{ }}k = 1,2 \), we can define a composition rule “⊙” for the U k by writing
$$ {U_3} = {\rm{ }}{U_1}{\rm{ }} \odot {\rm{ }}{U_2}, $$
$$ {U_3} = {\tilde P_3}({f_1}{f_2}). $$


Harmonic Function RIEMANN Surface Integral Operator Entire Function Complex Variable 
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Copyright information

© Springer-Verlag Berlin · Heidelberg 1969

Authors and Affiliations

  • Stefan Bergman
    • 1
  1. 1.Department of MathematicsStanford UniversityStanfordUSA

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