Science Journal of Applied Mathematics and Statistics

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Conversely Convergence Theorem of Fabry Gap

Received: 29 May 2015    Accepted: 10 June 2015    Published: 25 June 2015
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Abstract

Our previous paper conducted to prove a variation of the converse of Fabry Gap theorem concerning the location of singularities of Taylor-Dirichlet series, on the boundary of convergence. In the present paper, we prove conversely convergence theorem of Fabry Gap. This is another proof of Fabry Gap theorem. This prove may be of interest in itself.

DOI 10.11648/j.sjams.20150304.12
Published in Science Journal of Applied Mathematics and Statistics (Volume 3, Issue 4, August 2015)
Page(s) 177-183
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Dirichlet Series, Entire Functions, Fabry Gap Theorem

References
[1] Abbasi. N., Gorji, M., “On Convergence a Variation of the Converse of Fabry Gap Theorem, ” Science Journal of Applied Mathematics and Statistics, 3 (2), (2015), 58-62.
[2] Berenstein, C.A., and Gay Roger, “Complex Analysis and Special Topics in Harmonic Analysis” (New York, Inc: Springer-Verlag), (1995).
[3] Blambert, M. and Parvatham, R., “Ultraconvergence et singualarites pour une classe de series d exponentielles.” Universite de Grenoble. Annales de l’Institut Fourier, 29(1), (1979), 239–262.
[4] Blambert, M. and Parvatham, R., “Sur une inegalite fondamentale et les singualarites d une fonction analytique definie par un element LC-dirichletien. ” Universite de Grenoble. Annales de l’Institut Fourier, 33(4), (1983), 135–160.
[5] Berland, M., “On the convergenve and singularities of analytic functions defined by E-Dirichletian elements. ” Annales des Sciences Mathematiques du Quebec, 22(1),(1998), 1–15.
[6] Boas, R.P. Jr, , “Entire Functions,” (New York: Academic Press), (1954).
[7] Erdos, P., “Note on the converse of Fabry's Gap theorem,” Trans. Amer. Math. Soc., 57, (1945), 102-104.
[8] Polya, G., “On converse Gap theorems, ” Trans. Amer. Math. Soc., 52, (1942), 65-71.
[9] Levin, B. Ya., “Distribution of Zeros of Entire Functions,” (Providence, R.I.: Amer. Math. Soc.), (1964).
[10] Levin, B. Ya., “Lectures on Entire Functions, ” (Providence, R.I.: Amer. Math. Soc.), (1996).
[11] Levinson, N., “Gap and Density Theorems. ” American Mathematical Society Colloquium Publications, Vol. 26 (New York: Amer. Math. Soc.), (1940).
[12] Mandelbrojt, S., “Dirichlet Series, Principles and Methods,” (Dordrecht: D. Reidel Publishing Co.), (1972), pp. x‏166.
[13] Valiron, M. G., “Sur les solutions des equations differentielles lineaires d'ordre infni et a coeffcients constants, ” Ann.Ecole Norm Trans, 3, 46, (1929), 25-53.
[14] Zikkos, E., “On a theorem of Norman Levinson and a variation of the Fabry Gap theorem,” Complex Variables, 50 (4), (2005), 229-255.
Author Information
  • Department of Mathematics, Faculty of science, Lorestan University, Khoramabad, Islamic Republic of Iran

  • Department of Mathematics, Faculty of science, Lorestan University, Khoramabad, Islamic Republic of Iran

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    Naser Abbasi, Molood Gorji. (2015). Conversely Convergence Theorem of Fabry Gap. Science Journal of Applied Mathematics and Statistics, 3(4), 177-183. https://doi.org/10.11648/j.sjams.20150304.12

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    ACS Style

    Naser Abbasi; Molood Gorji. Conversely Convergence Theorem of Fabry Gap. Sci. J. Appl. Math. Stat. 2015, 3(4), 177-183. doi: 10.11648/j.sjams.20150304.12

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    AMA Style

    Naser Abbasi, Molood Gorji. Conversely Convergence Theorem of Fabry Gap. Sci J Appl Math Stat. 2015;3(4):177-183. doi: 10.11648/j.sjams.20150304.12

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  • @article{10.11648/j.sjams.20150304.12,
      author = {Naser Abbasi and Molood Gorji},
      title = {Conversely Convergence Theorem of Fabry Gap},
      journal = {Science Journal of Applied Mathematics and Statistics},
      volume = {3},
      number = {4},
      pages = {177-183},
      doi = {10.11648/j.sjams.20150304.12},
      url = {https://doi.org/10.11648/j.sjams.20150304.12},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.sjams.20150304.12},
      abstract = {Our previous paper conducted to prove a variation of the converse of Fabry Gap theorem concerning the location of singularities of Taylor-Dirichlet series, on the boundary of convergence. In the present paper, we prove conversely convergence theorem of Fabry Gap. This is another proof of Fabry Gap theorem. This prove may be of interest in itself.},
     year = {2015}
    }
    

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