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Domains of regularity of a harmonic function determined by the coefficients of its power series

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Summary

Theorems are proved which give a sufficient condition on the coefficients of a harmonic function in order that the function be regular in the domain {(x, y, z) | x < A } and is singular at the point (A, 0, 0).

Bibliography

  1. [1]

    Bergman, S., Zur theorie der ein-und mehrwertigen harmonischen funktionen des dreidimensional raumes, « Math. S. », 24: 641–699, 1926.

  2. [2]

    —— ——, Zur theorie der algebraischen potential funktionen des dreidimensional raumes, « Math. Annalen », 99: 629–659, 1928: 101: 634–558, 1929.

  3. [3]

    —— ——,On solutions with algebraic character of linear partial differential equations, « Trans. Math. Soc »., 68: 461–507, 1950.

  4. [4]

    —— ——,Essential singularities of solutions of a class of linear partial differential equations in three variables, « J. Rational Mach. and Anal. », 3: 539–560, 1954.

  5. [5]

    -- --,Integral Operators in the Theory of Linear Partial Differential Equations, « Ergebnisse der Math. N. F. » 23, Springer, 1961.

  6. [6]

    Bergman, S.,Some properties of a harmonic function of three variables given by its series development, « Arh. Rational Mech. and Anal. », 8: 207–222, 1962.

  7. [7]

    Bochner, S., andMartin, W.,Several Complex Variables, Princeton, « Princeton University Press », 1948.

  8. [8]

    Cima, J., «Some properties of harmonic functions generated by the Bergman-Whittaker operator, « Annali di Matematica pura ed applicata » (IV), vol. LXIII: 175–200, 1963.

  9. [9]

    Dienes, P.,The Taylor Series, New York, Dover, 1957.

  10. [10]

    Fuks, B. A.,Inlrodnction to the Theory of Analytic Functions of Several Complex Variables, « Amer. Math. Soc. », Providence, R. I., 1963.

  11. [11]

    Kreyszig, E.,On regular and singular harmonic functions of three real variables, « Arch. Rational Mech. and Anal., 4: 352–370, 1961.

  12. [12]

    Mitchell, J.,Some properties of solutions of partial differential equations given by their series development, « Duke Math. J. », 13: 87–107, 1946.

  13. [13]

    Mitchell, J.,Integrall theorems for harmonic vectors in three variables, « Math. Zeitscher. » 82: 314–334, 1963.

  14. [14]

    Whittaker, E. T.,On the partial differential equations of mathematical physics, « Math. Annmlen », 57: 333–335, 1903.

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This research was supported in part by the Air Force Office of Scientific Research.

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Cover, A.S. Domains of regularity of a harmonic function determined by the coefficients of its power series. Annali di Matematica 72, 11–27 (1966). https://doi.org/10.1007/BF02414324

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Keywords

  • Power Series
  • Harmonic Function