Surveys in Differential Geometry
Volume 9 (2004)
Discrete analytic functions: An exposition
Pages: 241 – 273
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.