An Integral Identity Relating Diamond and Square Domains

TL;DR

This paper derives an integral identity linking integrals over diamond-shaped and square domains in R^2, simplifying calculations by reducing diamond integrals to rectangular ones.

Contribution

It introduces a novel integral identity that relates integrals over diamond and square regions, facilitating easier computation of diamond domain integrals.

Findings

Integral over diamond region equals half of the integral over the square region.

The identity applies to functions invariant under discrete diagonal translations.

Simplifies calculations involving diamond-shaped domains in R^2.

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.