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Some properties of a harmonic function of three variables given by its series development

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References

  1. [1]

    Bergman, S.: Zur Theorie der ein und mehrwertigen harmonischen Funktionen des dreidimensionalen Raumes. Math. Z. 24, 641–669 (1926).

  2. [2]

    Bergman, S.: Zur Theorie der algebraischen Potentialfunktionen des dreidimensionalen Raumes. Math. Annalen 99, 629–659 (1928); 101, 534–558 (1929).

  3. [3]

    Bergman, S.: On solutions with algebraic character of linear partial differential equations. Trans. Amer. Math. Soc. 68, 461–507 (1950).

  4. [4]

    Bergman, S.: The coefficient problem in the theory of linear partial differential equations. Trans. Amer. Math. Soc. 73, 1–34 (1952).

  5. [5]

    Bergman, S.: Essential singularities of a class of linear partial differential equations in three variables. J. Rational Mech. Analysis 3, 539–560 (1954).

  6. [6]

    Bergman, S.: Integral Operators in the Theory of Linear Partial Differential Equations. Ergebnisse der Mathematik, N.F. 23, (1961).

  7. [7]

    Bieberbach, L.: Lehrbuch der Funktionentheorie, vol. II. Leipzig: B. G. Teubner 1931.

  8. [8]

    Dienes, P.: The Taylor Series. (Oxford University Press). Dover Edition, 1957, New York 1957.

  9. [9]

    Erdélyi, A.: Singularities of generalized axially symmetric potentials. Comm. Pure Appl. Math. 9, 403–414 (1956).

  10. [10]

    Gilbert, R. P.: Singularities of three-dimensional harmonic functions. Pacific J. Math. 10, 1243–1255 (1961).

  11. [11]

    Hadamard, J.: Essai sur l'étude des fonctions données par leur developpement de Taylor. J. de mathématiques pures et appliquées 8 (IV), 101–186 (1892).

  12. [12]

    Kreyszig, E.: Zur Behandlung elliptischer partieller Differentialgleichungen mit funktionentheoretischen Methoden. Z. angew. Math. und Mech. 40, 334–342 (1960).

  13. [13]

    Kreyszig, E.: On regular and singular harmonic functions of three variables. Arch. Rational Mech. Anal. 4, 352–370 (1960).

  14. [14]

    Madelung, E.: Die mathematischen Hilfsmittel des Physikers. Grundlagen der mathematischen Wissenschaften 4. Berlin: Springer 1935.

  15. [15]

    Mitchell, J.: Some properties of solutions of partial differential equations given by their series development. Duke Math. J. 13, 87–104 (1946).

  16. [16]

    Nevanlinna, R.: Le Théorème de Picard-Borel et la Théorie des Fonctions Meromorphes. Paris: Gauthier-Villars & Co. 1929.

  17. [17]

    White, A.: Singularities of harmonic functions of three variables generated by Whittaker-Bergman operators. Annales Polonici Mathematici 10, 81–100 (1961).

  18. [18]

    Whittaker, E. T.: On the partial differential equations of mathematical physics. Math. Annal. 57, 333–355 (1903).

  19. [19]

    Whittaker, E. T., & G. N. Watson: A Course of Modern Analysis. Cambridge: Cambridge University Press.

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Communicated by R. Finn

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Bergman, S. Some properties of a harmonic function of three variables given by its series development. Arch. Rational Mech. Anal. 8, 207 (1961). https://doi.org/10.1007/BF00277438

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Keywords

  • Harmonic Function
  • Entire Function
  • Meromorphic Function
  • Spherical Harmonic
  • Toeplitz Matrix