Discrete complex analysis on planar Quad-graphs

dc.contributor.authorBobenko, Alexander I.
dc.contributor.authorGünther, Felix
dc.date.accessioned2017-08-29T07:24:58Z
dc.date.available2017-08-29T07:24:58Z
dc.date.issued2016
dc.description.abstractWe develop further a linear theory of discrete complex analysis on general quad-graphs, extending previous work of Duffin, Mercat, Kenyon, Chelkak and Smirnov on discrete complex analysis on rhombic quad-graphs. Our approach based on the medial graph leads to generalizations as well as to new proofs of previously known discrete analogs of classical theorems. New results include in particular discretizations of Green’s first identity and Cauchy’s integral formula for the derivative of a holomorphic function. Another contribution is a discussion on the product of discrete holomorphic functions that is itself discrete holomorphic in a specific sense. In this paper, we focus on planar quad-graphs, but many notions and theorems can be easily adapted to discrete Riemann surfaces. In the case of planar parallelogram-graphs with bounded interior angles and bounded ratio of side lengths explicit formulae for a discrete Green’s function and discrete Cauchy’s kernels are obtained. This slightly generalizes the previous results on rhombic lattices. When we further restrict to the integer lattice of a two-dimensional skew coordinate system a discrete Cauchy’s integral formulae for higher order derivatives is derived.en
dc.identifier.isbn978-3-662-50447-5
dc.identifier.urihttps://depositonce.tu-berlin.de/handle/11303/6668
dc.identifier.urihttp://dx.doi.org/10.14279/depositonce-6109
dc.language.isoen
dc.relation.ispartof10.1007/978-3-662-50447-5
dc.rights.urihttps://creativecommons.org/licenses/by-nc/2.5/
dc.subject.ddc510 Mathematik
dc.subject.otherdiscrete complex analysisen
dc.subject.otherquad-graphen
dc.subject.othergreen’s functionen
dc.subject.othercauchy’s integral formulaeen
dc.subject.otherparallelogram-graphen
dc.titleDiscrete complex analysis on planar Quad-graphsen
dc.typeBook Part
dc.type.versionpublishedVersion
dcterms.bibliographicCitation.booktitleAdvances in discrete differential geometry
dcterms.bibliographicCitation.doi10.1007/978-3-662-50447-5_2
dcterms.bibliographicCitation.editorBobenko, Alexander I.
dcterms.bibliographicCitation.originalpublishernameSpringer
dcterms.bibliographicCitation.originalpublisherplaceBerlin, Heidelberg
dcterms.bibliographicCitation.pageend132
dcterms.bibliographicCitation.pagestart57
tub.accessrights.dnbfree
tub.affiliationFak. 2 Mathematik und Naturwissenschaften::Inst. Mathematik::FG Geometrie und Integrable Systemede
tub.affiliation.facultyFak. 2 Mathematik und Naturwissenschaftende
tub.affiliation.groupFG Geometrie und Integrable Systemede
tub.affiliation.instituteInst. Mathematikde
tub.publisher.universityorinstitutionTechnische Universität Berlin

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