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The Bergman kernel function:
Explicit formulas and zeroes


Authors: Harold P. Boas, Siqi Fu and Emil J. Straube
Journal: Proc. Amer. Math. Soc. 127 (1999), 805-811
MSC (1991): Primary 32H10
DOI: https://doi.org/10.1090/S0002-9939-99-04570-0
MathSciNet review: 1469401
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Abstract | References | Similar Articles | Additional Information

Abstract: We show how to compute the Bergman kernel functions of some special domains in a simple way. As an application of the explicit formulas, we show that the Bergman kernel functions of some convex domains, for instance the domain in \begin{math}\mathbb{C}^3\end{math} defined by the inequality \begin{math}|z_1|+|z_2|+|z_3|<1\end{math}, have zeroes.


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  • 1. Steven R. Bell, The Bergman kernel function and proper holomorphic mappings, Trans. Amer. Math. Soc. 270 (1982), no. 2, 685-691. MR 83i:32033
  • 2. Stefan Bergmann (Bergman), Zur Theorie von pseudokonformen Abbildungen, Mat. Sb. (N.S.) 1 (43) (1936), no. 1, 79-96.
  • 3. Harold P. Boas, The Lu Qi-Keng conjecture fails generically, Proc. Amer. Math. Soc. 124 (1996), no. 7, 2021-2027. MR 96i:32024
  • 4. Bruce L. Chalmers, On boundary behavior of the Bergman kernel function and related domain functionals, Pacific J. Math. 29 (1969), 243-250. MR 40:402
  • 5. John P. D'Angelo, A note on the Bergman kernel, Duke Math. J. 45 (1978), no. 2, 259-265. MR 57:12906
  • 6. -, An explicit computation of the Bergman kernel function, J. Geometric Analysis 4 (1994), no. 1, 23-34. MR 95a:32039
  • 7. G. P. Egorychev, Integral representation and the computation of combinatorial sums, Translations of Mathematical Monographs, vol. 59, American Mathematical Society, 1984. MR 85a:05008
  • 8. Gábor Francsics and Nicholas Hanges, The Bergman kernel of complex ovals and multivariable hypergeometric functions, J. Funct. Anal. 142 (1996), no. 2, 494-510. MR 97m:32039
  • 9. -, Asymptotic behavior of the Bergman kernel and hypergeometric functions, Multidimensional Complex Analysis and Partial Differential Equations, Contemporary Mathematics, vol. 205, American Mathematical Society, 1997, pp. 79-92. CMP 97:12
  • 10. Marek Jarnicki and Peter Pflug, Invariant Distances and Metrics in Complex Analysis, de Gruyter, 1993. MR 94k:32039
  • 11. Ewa Ligocka, On the Forelli-Rudin construction and weighted Bergman projections, Studia Math. 94 (1989), no. 3, 257-272. MR 90i:32034
  • 12. Qi-Keng Lu (K. H. Look), On Kaehler manifolds with constant curvature, Chinese Math. 8 (1966), 283-298. MR 34:6806
  • 13. K. Oeljeklaus, P. Pflug, and E. H. Youssfi, The Bergman kernel of the minimal ball and applications, Ann. Inst. Fourier (Grenoble) 47 (1997), no. 3, 915-928. MR 98d:32028
  • 14. I. Ramadanov, Sur une propriété de la fonction de Bergman, C. R. Acad. Bulgare Sci. 20 (1967), 759-762. MR 37:1632
  • 15. B. S. Zinov$'$ev, On reproducing kernels for multicircular domains of holomorphy, Siberian Math. J. 15 (1974), 24-33. MR 48:11555

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Additional Information

Harold P. Boas
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843–3368
Email: boas@math.tamu.edu

Siqi Fu
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843–3368
Address at time of publication: Department of Mathematics, University of Wyoming, Laramie, Wyoming 82071-3036
Email: sfu@math.tamu.edu

Emil J. Straube
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843–3368
Email: straube@math.tamu.edu

DOI: https://doi.org/10.1090/S0002-9939-99-04570-0
Received by editor(s): June 30, 1997
Additional Notes: This research was supported in part by NSF grant number DMS 9500916.
Communicated by: Steven R. Bell
Article copyright: © Copyright 1999 American Mathematical Society

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