The-orem2.16 shows that the discrete outer derivative is a derivative of the discrete wedge product. Furthermore, discrete Cauchy integral formulas for discrete holomorphic functions are derived in Theorem 2.35 and for the discrete derivative of a discrete holomorphic function at the vertices of the quadrilateral graph in Theorem 2.36.

## Discrete Derivatives of Functions on the Faces of Λ

Therefore, the coefficient before f(b+)− f(b−) on the right side of the first equation in the lemma is equal. Note Note that in their work on discrete complex analysis on rhombic quadragraphs, Kenyon [16] and Mercat [20] did not give explicit formulas for discrete derivatives, but instead defined −∂♦ and −¯∂♦ as formal adjoints. of discrete derivatives ∂¯Λ and ∂Λ, respectively.

## Discrete Exterior Calculus

### Discrete Exterior Derivative

Similarly, f hd zi is closed, so f ·is discrete holomorph in the sense that a discrete Morera's theorem is true. Then the product f ·g is not discretely holomorphic at allQ∈V(♦) (viewed as vertices of the quad graph described above) where∂Λg(Q)=0.

### Discrete Wedge Product

Starting with discrete uniforms of type♦defined on the edges of X, a discrete wedge product is defined on (half of) the faces of X. Definition Letω= pd z+qdz¯enω=pd z+qd¯zbe two discrete uniforms of type♦defined on the oriented edges of X0.

### Discrete Hodge Star

Actually, we could define a discrete wedge product of two discrete one-forms of type Λ in much the same way, giving us a discrete two-form of type Λ. Also, the discrete Hodge star of a discrete one form could not be defined only for those of type♦ in the next section. These theorems imply that both the discrete wedge product and the discrete Hodge star of discrete ones of type♦ can be defined in a map-independent manner.

In fact, a discrete one-form of typeΛ cannot be canonically defined on a discrete Riemann surface, unlike discrete one-form of type♦. Since our interest is therefore in the latter, we do not define a discrete wedge product or a discrete Hodge star for discrete one-forms of typeΛ.

## Discrete Laplacian

### Definition and Basic Properties

The following factorization of the discrete Laplacian in terms of discrete derivatives generalizes the corresponding result given by Chelkak and Smirnov in [6] to general quad graphs. Proof Since the definitions of the discrete Hodge star and the discrete outer derivative mimic the classical theory and ∂♦∂¯Λf(v)= ¯∂♦∂Λf(v) by Corollary2.11,. Indeed, ifes denotes an edge of X parallel to the black diagonalvvsandes∗an edge parallel to the double diagonal, then. use discrete Stokes' Theorem2.9in the first and third equality, Proposition2.18 which compares the integration of the discrete Hodge star of a discrete one-form of type♦with the integration of the discrete one-form f itself in the second equality, and| ρQs| = |e∗s|/|es|for the last step.

Proof (i) Since the discrete outer derivative is a derivative for the discrete wedge product of Theorem 2.16, . Note that the existence of a real function f˜such that f +if˜is discrete holomorphic already requires that f is discrete harmonic at all interior vertices ofΛ0 (i.e. V(Λ0)\V(∂Λ0)) due by corollary2 .21(ii) which implies that the real part of a discrete holomorphic function is discrete harmonic. i) The discrete harmonic conjugate f is unique up to two additive real constants onΓ0andΓ0∗. ii) If ♦0 forms a single connected closed region, then there exists a discrete harmonic conjugate f.˜.

### Discrete Dirichlet Energy

The same argument holds for general quad graphs with the discrete Dirichlet energy defined here. Using a different notation, Skopenkov also proved the existence and uniqueness of solutions of the discrete Dirichlet boundary value problem [23]. In particular, the solution of the discrete Dirichlet boundary value problem is given by the unique minimizer of E♦0.

It follows that precisely the minima of E♦0 are discrete harmonics and therefore solve the discrete Dirichlet boundary value problem. In particular, the discrete Dirichlet energy of f is bounded by zero if a domain contains F.

## Discrete Green’s Functions

Remark It is important to note that the discrete Green's functions as we defined them are by no means unique. So it might be more convenient to call these functions functions of the discrete Green's function type, but for the sake of convenience, we still call them discrete Green's functions. However, when considering planar parallelogram graphs with bounded interior angles and bounded ratio of side lengths in Section 3.3, the existence of a discrete Green's function with asymptotics generalizing the corresponding result for rhombic quadratics is proved.

The following notion of discrete Green's functions in a discrete domain follows the introduction of Chelkak and Smirnov in [6]. Since the difference of two discrete Green's functions inΛ0forv0 is discrete harmonic in V(Λ0) and equal to zero in the limit V(∂Λ0), it must be identically zero by Lemma2.28 since the zero function is the unique solution of the corresponding discrete Dirichlet limit. the problem of value.

## Discrete Cauchy’s Integral Formulae

### A Different Notation

For Q∈W♦ we denote its white vertices by w−(Q),w+(Q), so that the correspondingly oriented diagonal goes fromw−(Q)tow+(Q). For Q∈ B♦ we denote its black vertices by b−(Q),b+(Q), so that the corresponding oriented diagonal goes from b−(Q)tob+(Q). Between the closed paths B and W there is a cycle P on the medial graph X that includes exactly all edges [Q,v]with Q∈W♦ andv∈B incident to Q and all edges [Q,v]with Q∈ B ♦ and v∈W event to Q.

The above Lemma2.37 directly relates this formulation to the one of the discrete Cauchy's integral formula in Theorem 2.35. System 2.38 Let W be a closed cycle on Γ∗ and B be the corresponding closed cycle on Γ as above, and let f,g:V(Λ)→C.

## 3 Discrete Complex Analysis on Planar Parallelogram-Graphs

### Preliminaries

To focus on the computation of the asymptotics of a given discrete Green function and discrete Cauchy kernels, we defer to the appendix the discussion of some necessary combinatorial and geometric results on planar parallelogram graphs. The two conditions we use are that all interior angles of the parallelograms are bounded (the same condition was used by Chelkak and Smirnov in [6]) and that the ratio of the side lengths of the parallelograms is also bounded. In the rest of the article, we use the following shorthand notation. i) Choose an arbitrary directed path from edgese1,. which does not depend on the choice of the path from vtovd to Lemma3.1. ii) Choose a vertex vQ incident on Q and a directed path from edges e1,.

Note that J(Q,v) does not depend on the choice of path from v to vQ by Lemma3.1 nor on the choice of vQ with a similar argument as in the proof of the above lemma. Finally, the notion of the argument of a complex number will become important in the continuation.

### Discrete Exponential Function

Remark If all parallelograms are rhombuses with side length one, we have J(x,x)=x−x. In our paper, it will usually be an arbitrary real number rather than a numeric modulus 2π. Definition For a complex number λ=0, a real number φ such that λ/exp(iφ) is a positive real number is called an argument of λ.

On the other hand, arg(λ) denotes the unique argumentφofλ contained in the interval(−π, π].

### Asymptotics of the Discrete Green’s Function

In particular, the real part of the discrete logarithm log(·;v0) is a well-defined function on V(Λ). For this, we only need to show that the branches of the logarithm are well defined. All the claims in the first three parts hold (essentially) the same under complex plane rotation, so we can assume that θ0= −(π−α0)/2.

It is easy to check that the claim is invariant under rotation of the complex plane, so we can assume θ0= −(π−α0)/2. In total, the asymptotics for the real part of the integration along two circles.

### Asymptotics of Discrete Cauchy’s Kernels

Therefore we get for the discrete derivative of 8πG(·;v0) at faceQ:. and we have used the identity. iii) Let be the difference between two discrete Cauchy kernels with respect to tov0. Since we do not have discrete Green's functions on V(♦), we have to derive a discrete Cauchy kernel on V(Λ) differently. For the discrete Cauchy kernel KQ0 given in Proposition 3.9, ∂ΛKQ0 has the following asymptotics as|v| → ∞:. iv).

Up to two additional complex constants in Γ and Γ∗, there are no further discrete Cauchy kernels with respect to Q0 such that its discrete derivative has asymptotes o(|Q−Q0|−1/2) as|Q|. Remarks Note that Kenyon [16] and Chelkak and Smirnov [6] proved in the rhombic setting the stronger result that there exists a unique discrete Cauchy kernel in V(Λ) with respect to Q0 with asymptoticso(1)as|v|.

### Integer Lattice

Moreover, the discrete derivatives viewed as the dual of Λ coincide with the discrete derivatives viewed as a parallelogram graph. Moreover, the discrete Laplacian is now defined in the same way for functions on V(Λ) and functions on V(♦). Our main interests lie in giving discrete Cauchy's integral formulas for higher order derivatives of a discrete holomorphic function and determining the asymptotics of higher order discrete derivatives of the discrete Cauchy's kernel given in Section 3.4.

For any discrete counterclockwise oriented contour Cx0 in the medial graph X that includes all points x∈V(Λ)∪V(♦)with D(x,x0)n/2,. ii) If KQ0 is the discrete Cauchy kernel given in Proposition 3.9, then (−1)n. From the assumptions on Cx0, a discrete form∂∂¯ n−1Kx0dz¯ vanishes in Cx0. by Stokes' discrete Theorem 2.9 in the second equation and Theorem 2.16 and Proposition 2.10 stating that is a derivative for the discrete wedge product and dd f =0 in the third equation.

## Appendix: Planar Parallelogram-Graphs

Without loss of generality, we assume that the planes of Sare have continued in such a way that the projections of the corresponding coordinates are strictly increasing. So there is a vertex that is such that the argument of the directed edge vertex in[arg(e),arg(e)+π−α0]. This contradicts the monotonicity of SQj j−1,j in the direction i aSj, because the radius x+taSj,t >0 is not in the interior of the cone above.

If Q=Qm−1.0, then we look at the semi-infinite part of the strip S˜0, which is different from S0, passing through Q and going inside the upper bounded region. Sm−1 such that Q lies on one of the strips, say Sk, and the intersection of S0 and Sm−1 is Q.

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