Discrete complex analysis – the medial graph approach

Alexander I. Bobenko[1]; Felix Günther[2]

  • [1] Institut für Mathematik MA 8-4 Technische Universität Berlin Straße des 17. Juni 136 10623 BERLIN GERMANY
  • [2] Institut für Mathematik MA 8-3 Technische Universität Berlin Straße des 17. Juni 136 10623 BERLIN GERMANY

Actes des rencontres du CIRM (2013)

  • Volume: 3, Issue: 1, page 159-169
  • ISSN: 2105-0597

Abstract

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We discuss a new formulation of the linear theory of discrete complex analysis on planar quad-graphs based on their medial graphs. It generalizes the theory on rhombic quad-graphs developed by Duffin, Mercat, Kenyon, Chelkak and Smirnov and follows the approach on general quad-graphs proposed by Mercat. We provide discrete counterparts of the most fundamental objects in complex analysis such as holomorphic functions, differential forms, derivatives, and the Laplacian. Also, we discuss discrete versions of important fundamental theorems such as Green’s identities and Cauchy’s integral formulae. For the first time, Green’s first identity and Cauchy’s integral formula for the derivative of a holomorphic function are discretized.

How to cite

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Bobenko, Alexander I., and Günther, Felix. "Discrete complex analysis – the medial graph approach." Actes des rencontres du CIRM 3.1 (2013): 159-169. <http://eudml.org/doc/275333>.

@article{Bobenko2013,
abstract = {We discuss a new formulation of the linear theory of discrete complex analysis on planar quad-graphs based on their medial graphs. It generalizes the theory on rhombic quad-graphs developed by Duffin, Mercat, Kenyon, Chelkak and Smirnov and follows the approach on general quad-graphs proposed by Mercat. We provide discrete counterparts of the most fundamental objects in complex analysis such as holomorphic functions, differential forms, derivatives, and the Laplacian. Also, we discuss discrete versions of important fundamental theorems such as Green’s identities and Cauchy’s integral formulae. For the first time, Green’s first identity and Cauchy’s integral formula for the derivative of a holomorphic function are discretized.},
affiliation = {Institut für Mathematik MA 8-4 Technische Universität Berlin Straße des 17. Juni 136 10623 BERLIN GERMANY; Institut für Mathematik MA 8-3 Technische Universität Berlin Straße des 17. Juni 136 10623 BERLIN GERMANY},
author = {Bobenko, Alexander I., Günther, Felix},
journal = {Actes des rencontres du CIRM},
keywords = {Discrete complex analysis; quad-graphs; medial graph; Green’s identities; Cauchy’s integral formulae},
language = {eng},
month = {11},
number = {1},
pages = {159-169},
publisher = {CIRM},
title = {Discrete complex analysis – the medial graph approach},
url = {http://eudml.org/doc/275333},
volume = {3},
year = {2013},
}

TY - JOUR
AU - Bobenko, Alexander I.
AU - Günther, Felix
TI - Discrete complex analysis – the medial graph approach
JO - Actes des rencontres du CIRM
DA - 2013/11//
PB - CIRM
VL - 3
IS - 1
SP - 159
EP - 169
AB - We discuss a new formulation of the linear theory of discrete complex analysis on planar quad-graphs based on their medial graphs. It generalizes the theory on rhombic quad-graphs developed by Duffin, Mercat, Kenyon, Chelkak and Smirnov and follows the approach on general quad-graphs proposed by Mercat. We provide discrete counterparts of the most fundamental objects in complex analysis such as holomorphic functions, differential forms, derivatives, and the Laplacian. Also, we discuss discrete versions of important fundamental theorems such as Green’s identities and Cauchy’s integral formulae. For the first time, Green’s first identity and Cauchy’s integral formula for the derivative of a holomorphic function are discretized.
LA - eng
KW - Discrete complex analysis; quad-graphs; medial graph; Green’s identities; Cauchy’s integral formulae
UR - http://eudml.org/doc/275333
ER -

References

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  13. J. Lelong-Ferrand, Représentation conforme et transformations à intégrale de Dirichlet bornée, (1955), Gauthier-Villars, Paris Zbl0064.32204MR69895
  14. C. Mercat, Discrete Riemann surfaces and the Ising model, Commun. Math. Phys. 218 (2001), 177-216 Zbl1043.82005MR1824204
  15. C. Mercat, Discrete Riemann surfaces, Handbook of Teichmüller theory. Vol. I 11 (2007), 541-575, Eur. Math. Soc., Zurich Zbl1136.30315MR2349680
  16. C. Mercat, Discrete complex structure on surfel surfaces, Proceedings of the 14th IAPR international conference on Discrete geometry for computer imagery (2008), 153-164, Springer-Verlag, Berlin, Heidelberg Zbl1138.68607MR2503463
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