Discrete Complex Analysis on Planar Quad-Graphs

Alexander I. Bobenko () and Felix Günther ()
Additional contact information
Alexander I. Bobenko: Technische Universität Berlin, Inst. für Mathematik
Felix Günther: Technische Universität Berlin, Inst. für Mathematik

A chapter in Advances in Discrete Differential Geometry, 2016, pp 57-132 from Springer

Abstract: Abstract We 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.

Keywords: Discrete complex analysis; Quad-graph; Green’s function; Cauchy’s integral formulae; Parallelogram-graph (search for similar items in EconPapers)
Date: 2016
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-662-50447-5_2

Ordering information: This item can be ordered from
http://www.springer.com/9783662504475

DOI: 10.1007/978-3-662-50447-5_2

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().