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An Integral Identity Relating Diamond and Square Domains

Agustín Domínguez-Cruz

Abstract

We establish an integral identity for functions on R^2 that are invariant under discrete diagonal translations. The identity shows that integration over the diamond-shaped region |x| + |y| <= L is exactly one half of the integral over the square domain [-L, L]^2, allowing diamond-domain integrals to be reduced to easier rectangular integrations.

An Integral Identity Relating Diamond and Square Domains

Abstract

We establish an integral identity for functions on R^2 that are invariant under discrete diagonal translations. The identity shows that integration over the diamond-shaped region |x| + |y| <= L is exactly one half of the integral over the square domain [-L, L]^2, allowing diamond-domain integrals to be reduced to easier rectangular integrations.
Paper Structure (2 sections, 1 theorem, 12 equations, 1 figure)

This paper contains 2 sections, 1 theorem, 12 equations, 1 figure.

Table of Contents

  1. Main result
  2. Example

Key Result

Theorem 1

Let $f:\mathbb{R}^2\to\mathbb{C}$ satisfy the translation invariances for all $(x,y)\in\mathbb{R}^2$. Then

Figures (1)

  • Figure 1: Visualization of the integrand in Eq. \ref{['example']} on the domains $S=[-\pi,\pi]^2$, $D=\{(u,v):|u|+|v|\le\pi\}$, and $S\setminus D$, for parameters $A=1$, $B=0.5$, $C=-0.8$, and $D=0.2$.

Theorems & Definitions (2)

  • Theorem 1
  • proof