A general form of Green’s Formula and the Cauchy Integral Theorem

Cufí, Julià; Verdera, Joan

doi:10.1090/S0002-9939-2014-12418-X

A general form of Green’s Formula and the Cauchy Integral Theorem
HTML articles powered by AMS MathViewer

by Julià Cufí and Joan Verdera PDF

Proc. Amer. Math. Soc. 143 (2015), 2091-2102 Request permission

Abstract:

We prove a general form of Green’s Formula and the Cauchy Integral Theorem for arbitrary closed rectifiable curves in the plane. We use Vituśkin’s localization of singularities method and a decomposition of a rectifiable curve in terms of a sequence of Jordan rectifiable sub-curves due to Carmona and Cufí.

References

Robert B. Burckel, An introduction to classical complex analysis. Vol. 1, Pure and Applied Mathematics, vol. 82, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR 555733, DOI 10.1007/978-3-0348-9374-9
Joan Josep Carmona and Julià Cufí, The index of a plane curve and Green’s formula, Rend. Circ. Mat. Palermo (2) 53 (2004), no. 1, 103–128. MR 2021679, DOI 10.1007/BF02921431
Joan Josep Carmona and Julià Cufí, The calculation of the $L^2$-norm of the index of a plane curve and related formulas, J. Anal. Math. 120 (2013), 225–253. MR 3095153, DOI 10.1007/s11854-013-0019-9
Paul J. Cohen, On Green’s theorem, Proc. Amer. Math. Soc. 10 (1959), 109–112. MR 104249, DOI 10.1090/S0002-9939-1959-0104249-3
Robert M. Fesq, Green’s formula, linear continuity, and Hausdorff measure, Trans. Amer. Math. Soc. 118 (1965), 105–112. MR 174696, DOI 10.1090/S0002-9947-1965-0174696-X
Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969. MR 0410387
J. H. Michael, An approximation to a rectifiable plane curve, J. London Math. Soc. 30 (1955), 1–11. MR 66445, DOI 10.1112/jlms/s1-30.1.1
I. P. Natanson, Theory of Functions of a real variable, Vol. 1, Frederich Ungar Publ., New York, 1961.
Georg Nöbeling, Eine allgemeine Fassung des Hauptsatzes der Funktionentheorie von Cauchy, Math. Ann. 121 (1949), 54–66 (German). MR 29976, DOI 10.1007/BF01329616
Joan Verdera, Removability, capacity and approximation, Complex potential theory (Montreal, PQ, 1993) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 439, Kluwer Acad. Publ., Dordrecht, 1994, pp. 419–473. MR 1332967
A. G. Vituškin, Analytic capacity of sets in problems of approximation theory, Uspehi Mat. Nauk 22 (1967), no. 6 (138), 141–199 (Russian). MR 0229838