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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  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. 583 (2005), 117-161 Zbl1099.37054MR2146854
  2. U. Bücking, Approximation of conformal mappings by circle patterns, Geom. Dedicata 137 (2008), 163-197 Zbl1151.52018MR2449151
  3. D. Chelkak, S. Smirnov, Discrete complex analysis on isoradial graphs, Adv. Math. 228 (2011), 1590-1630 Zbl1227.31011MR2824564
  4. D. Chelkak, S. Smirnov, Universality in the 2D Ising model and conformal invariance of fermionic observables, Invent. Math. 189 (2012), 515-580 Zbl1257.82020MR2957303
  5. R. Courant, K. Friedrichs, H. Lewy, Über die partiellen Differentialgleichungen der mathematischen Physik, Math. Ann. 100 (1928), 32-74 Zbl54.0486.01MR1512478
  6. R.J. Duffin, Basic properties of discrete analytic functions, Duke Math. J. 23 (1956), 335-363 Zbl0070.30503MR78441
  7. R.J. Duffin, Potential theory on a rhombic lattice, J. Comb. Th. 5 (1968), 258-272 Zbl0247.31003MR232005
  8. J. Ferrand, Fonctions préharmoniques et fonctions préholomorphes, Bull. Sci. Math. Ser. 2 68 (1944), 152-180 Zbl0063.01349MR13411
  9. F. Günther, Discrete Riemann surfaces and integrable systems, (2014) 
  10. R.Ph. Isaacs, A finite difference function theory, Univ. Nac. Tucumán. Rev. A 2 (1941), 177-201 Zbl0060.13009MR6391
  11. R. Kenyon, Conformal invariance of domino tiling, Ann. Probab. 28 (2002), 759-795 Zbl1043.52014MR1782431
  12. R. Kenyon, The Laplacian and Dirac operators on critical planar graphs, Invent. math. 150 (2002), 409-439 Zbl1038.58037MR1933589
  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
  17. B. Rodin, D. Sullivan, The convergence of circle packings to the Riemann mapping, J. Diff. Geom. 26 (1987), 349-360 Zbl0694.30006MR906396
  18. M. Skopenkov, The boundary value problem for discrete analytic functions, Adv. Math. 240 (2013), 61-87 Zbl1278.39008MR3046303
  19. H. Whitney, Product on complexes, Ann. Math. 39 (1938), 397-432 Zbl0019.14204MR1503416

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