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We generalize the linear theory of discrete complex analysis on rhombic quad-graphs developed by Duffin, Mercat, Kenyon, Chelkak and Smirnov to the case of general quad-graphs. Our main focus lies on discretizing the theory of complex analysis on the plane, but we look at discrete Riemann surfaces as well.

We provide discrete analogs of several classical results. Introducing integration on the medial graph allows us to formulate and prove the discrete Cauchy formula in an intuitive way, and to develop a discrete Cauchy formula for the derivative of a discrete holomorphic function.