A constructive Schwarz reflection principle
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- by Jeremy Clark PDF
- Trans. Amer. Math. Soc. 355 (2003), 4569-4579 Request permission
Abstract:
We prove a constructive version of the Schwarz reflection principle. Our proof techniques are in line with Bishop’s development of constructive analysis. The principle we prove enables us to reflect analytic functions in the real line, given that the imaginary part of the function converges to zero near the real line in a uniform fashion. This form of convergence to zero is classically equivalent to pointwise convergence, but may be a stronger condition from the constructivist point of view.References
- Errett Bishop and Douglas Bridges, Constructive analysis, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 279, Springer-Verlag, Berlin, 1985. MR 804042, DOI 10.1007/978-3-642-61667-9
- Douglas Bridges and Fred Richman, Varieties of constructive mathematics, London Mathematical Society Lecture Note Series, vol. 97, Cambridge University Press, Cambridge, 1987. MR 890955, DOI 10.1017/CBO9780511565663
- Theodore W. Gamelin, Complex analysis, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 2001. MR 1830078, DOI 10.1007/978-0-387-21607-2
Additional Information
- Jeremy Clark
- Affiliation: 107 Rue de Sèvres, Paris 75006, France
- Email: jclark@noos.fr
- Received by editor(s): November 5, 2002
- Received by editor(s) in revised form: November 11, 2002
- Published electronically: July 8, 2003
- © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 355 (2003), 4569-4579
- MSC (2000): Primary 03F60, 30E99
- DOI: https://doi.org/10.1090/S0002-9947-03-03359-2
- MathSciNet review: 1990762