Discrete complex analysis – the medial graph approach

Alexander I. Bobenko; Felix Günther

doi:10.5802/acirm.65

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, Volume 3 (2013) no. 1, pp. 159-169.

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.

Published online: 2014-11-12

Zbl: 06938613

DOI: 10.5802/acirm.65

Classification: 39A12, 30G25
Keywords: Discrete complex analysis, quad-graphs, medial graph, Green’s identities, Cauchy’s integral formulae

Author's affiliations:

Alexander I. Bobenko ¹; Felix Günther ²

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Alexander I. Bobenko; Felix Günther. Discrete complex analysis – the medial graph approach. Actes des rencontres du CIRM, Volume 3 (2013) no. 1, pp. 159-169. doi : 10.5802/acirm.65. https://acirm.centre-mersenne.org/articles/10.5802/acirm.65/

References
Cited by

[1] A.I. Bobenko; C. Mercat; Yu.B. Suris Linear and nonlinear theories of discrete analytic functions. Integrable structure and isomonodromic Green’s function, J. Reine Angew. Math., Volume 583 (2005), pp. 117-161 | DOI | MR

[2] U. Bücking Approximation of conformal mappings by circle patterns, Geom. Dedicata, Volume 137 (2008), pp. 163-197 | DOI | MR | Zbl

[3] D. Chelkak; S. Smirnov Discrete complex analysis on isoradial graphs, Adv. Math., Volume 228 (2011), pp. 1590-1630 | DOI | MR | Zbl

[4] D. Chelkak; S. Smirnov Universality in the 2D Ising model and conformal invariance of fermionic observables, Invent. Math., Volume 189 (2012) no. 3, pp. 515-580 | DOI | MR | Zbl

[5] R. Courant; K. Friedrichs; H. Lewy Über die partiellen Differentialgleichungen der mathematischen Physik, Math. Ann., Volume 100 (1928), pp. 32-74 | DOI | Zbl

[6] R.J. Duffin Basic properties of discrete analytic functions, Duke Math. J., Volume 23 (1956) no. 2, pp. 335-363 | DOI | MR

[7] R.J. Duffin Potential theory on a rhombic lattice, J. Comb. Th., Volume 5 (1968), pp. 258-272 | DOI | MR | Zbl

[8] J. Ferrand Fonctions préharmoniques et fonctions préholomorphes, Bull. Sci. Math. Ser. 2, Volume 68 (1944), pp. 152-180 | Zbl

[9] F. Günther Discrete Riemann surfaces and integrable systems, Technische Universität Berlin, September (2014) (Ph. D. Thesis)

[10] R.Ph. Isaacs A finite difference function theory, Univ. Nac. Tucumán. Rev. A, Volume 2 (1941), pp. 177-201 | MR

[11] R. Kenyon Conformal invariance of domino tiling, Ann. Probab., Volume 28 (2002) no. 2, pp. 759-795 | MR | Zbl

[12] R. Kenyon The Laplacian and Dirac operators on critical planar graphs, Invent. math., Volume 150 (2002), pp. 409-439 | DOI | MR | Zbl

[13] J. Lelong-Ferrand Représentation conforme et transformations à intégrale de Dirichlet bornée, Gauthier-Villars, Paris, 1955 | Zbl

[14] C. Mercat Discrete Riemann surfaces and the Ising model, Commun. Math. Phys., Volume 218 (2001) no. 1, pp. 177-216 | DOI | MR | Zbl

[15] C. Mercat Discrete Riemann surfaces, Handbook of Teichmüller theory. Vol. I (IRMA Lect. Math. Theor. Phys.), Volume 11 (2007), pp. 541-575 | DOI | MR | Zbl

[16] C. Mercat Discrete complex structure on surfel surfaces, Proceedings of the 14th IAPR international conference on Discrete geometry for computer imagery (DGCI’08) (2008), pp. 153-164 | DOI | MR | Zbl

[17] B. Rodin; D. Sullivan The convergence of circle packings to the Riemann mapping, J. Diff. Geom., Volume 26 (1987) no. 2, pp. 349-360 | MR | Zbl

[18] M. Skopenkov The boundary value problem for discrete analytic functions, Adv. Math., Volume 240 (2013), pp. 61-87 | DOI | MR | Zbl

[19] H. Whitney Product on complexes, Ann. Math., Volume 39 (1938) no. 2, pp. 397-432 | DOI | MR | Zbl

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