Surveys in Differential Geometry

Volume 9 (2004)

Discrete analytic functions: An exposition

Pages: 241 – 273

DOI: http://dx.doi.org/10.4310/SDG.2004.v9.n1.a7

Author

L. Lovász (Microsoft Research, One Microsoft Way, Redmond, WA 98052, U.S.A.)

Abstract

Harmonic and analytic functions have natural discrete analogues. Harmonic functions can be defined on every graph, while analytic functions (or, more precisely, holomorphic forms) can be defined on graphs embedded in orientable surfaces. Many important properties of the “true” harmonic and analytic functions can be carried over to the discrete setting.

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