On holomorphic continuation of functions along boundary sections

S. A. Imomkulov; J. U. Khujamov

Mathematica Bohemica (2005)

  • Volume: 130, Issue: 3, page 309-322
  • ISSN: 0862-7959

Abstract

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Let D ' n - 1 be a bounded domain of Lyapunov and f ( z ' , z n ) a holomorphic function in the cylinder D = D ' × U n and continuous on D ¯ . If for each fixed a ' in some set E D ' , with positive Lebesgue measure mes E > 0 , the function f ( a ' , z n ) of z n can be continued to a function holomorphic on the whole plane with the exception of some finite number (polar set) of singularities, then f ( z ' , z n ) can be holomorphically continued to ( D ' × ) S , where S is some analytic (closed pluripolar) subset of D ' × .

How to cite

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Imomkulov, S. A., and Khujamov, J. U.. "On holomorphic continuation of functions along boundary sections." Mathematica Bohemica 130.3 (2005): 309-322. <http://eudml.org/doc/249606>.

@article{Imomkulov2005,
abstract = {Let $D^\{\prime \} \subset \mathbb \{C\}^\{n-1\}$ be a bounded domain of Lyapunov and $f(z^\{\prime \},z_n)$ a holomorphic function in the cylinder $D=D^\{\prime \}\times U_n$ and continuous on $\bar\{D\}$. If for each fixed $a^\{\prime \}$ in some set $E\subset \partial D^\{\prime \}$, with positive Lebesgue measure $\text\{mes\}\,E>0$, the function $f(a^\{\prime \},z_n)$ of $z_n$ can be continued to a function holomorphic on the whole plane with the exception of some finite number (polar set) of singularities, then $f(z^\{\prime \},z_n)$ can be holomorphically continued to $(D^\{\prime \}\times \mathbb \{C\})\setminus S$, where $S$ is some analytic (closed pluripolar) subset of $D^\{\prime \}\times \mathbb \{C\}$.},
author = {Imomkulov, S. A., Khujamov, J. U.},
journal = {Mathematica Bohemica},
keywords = {holomorphic function; holomorphic continuation; pluripolar set; pseudoconcave set; Jacobi-Hartogs series; holomorphic function; pluripolar set; pseudoconcave set; Jacobi-Hartogs series},
language = {eng},
number = {3},
pages = {309-322},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On holomorphic continuation of functions along boundary sections},
url = {http://eudml.org/doc/249606},
volume = {130},
year = {2005},
}

TY - JOUR
AU - Imomkulov, S. A.
AU - Khujamov, J. U.
TI - On holomorphic continuation of functions along boundary sections
JO - Mathematica Bohemica
PY - 2005
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 130
IS - 3
SP - 309
EP - 322
AB - Let $D^{\prime } \subset \mathbb {C}^{n-1}$ be a bounded domain of Lyapunov and $f(z^{\prime },z_n)$ a holomorphic function in the cylinder $D=D^{\prime }\times U_n$ and continuous on $\bar{D}$. If for each fixed $a^{\prime }$ in some set $E\subset \partial D^{\prime }$, with positive Lebesgue measure $\text{mes}\,E>0$, the function $f(a^{\prime },z_n)$ of $z_n$ can be continued to a function holomorphic on the whole plane with the exception of some finite number (polar set) of singularities, then $f(z^{\prime },z_n)$ can be holomorphically continued to $(D^{\prime }\times \mathbb {C})\setminus S$, where $S$ is some analytic (closed pluripolar) subset of $D^{\prime }\times \mathbb {C}$.
LA - eng
KW - holomorphic function; holomorphic continuation; pluripolar set; pseudoconcave set; Jacobi-Hartogs series; holomorphic function; pluripolar set; pseudoconcave set; Jacobi-Hartogs series
UR - http://eudml.org/doc/249606
ER -

References

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  1. 10.1007/BF02392348, Acta. Math. 149 (1982), 1–40. (1982) MR0674165DOI10.1007/BF02392348
  2. A local condition for single-valuedness of analytic functions, Math. USSR Sb. 89 (1972), 148–164. (Russian) (1972) MR0322144
  3. 10.1007/BF01448415, Math. Ann. 62 (1906), 1–88. (1906) MR1511365DOI10.1007/BF01448415
  4. On holomorphic continuation of functions with special singularities in n , Akad. Nauk Armyan. SSR Dokl. 76 (1983), 13–17. (Russian) (1983) MR0704694
  5. Note sur les familles des fonctions analytiques multiformes ets, J. Sci. Hiroshima Univ., Ser. A 4 (1934), 93–98. (1934) 
  6. Boundary problems and various classes of harmonic and subharmonic functions defined in arbitrary domains, Math. USSR Sb. 6 (1939), 345–375. (Russian) (1939) 
  7. 10.1007/BF01175583, Math. Z. 53 (1950), 84–95. (1950) MR0037365DOI10.1007/BF01175583
  8. Rational approximations and pluripolar sets, Math. USSR Sb. 119 (1982), 96–118. (1982) Zbl0511.32011MR0672412
  9. A criterion for rapid rational approximation in n , Math. USSR Sb. 125 (1984), 269–279. (Russian) (1984) MR0764481
  10. On continuation of functions with polar singularities, Mat. Sb., N. Ser. 132 (1987), 383–390. (Russian) (1987) MR0889599
  11. Introduction to the complex analysis. Part II, Nauka, Moskva, 1985. (Russian) (1985) MR0831938
  12. 10.1090/S0002-9939-1981-0593466-6, Proc. Amer. Math. Soc. 81 (1981), 243–249. (1981) Zbl0407.46046MR0593466DOI10.1090/S0002-9939-1981-0593466-6
  13. Continuation of functions along a fixed direction, Sibir. Math. Journal 229 (1988), 142–147. (Russian) (1988) 

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