# Box Integrals

### Description

By a "box integral" we mean here an expectation $\langle|\vec r - \vec q|^s \rangle$ where $\vec r$runs over the unit $n$-cube,with $\vec q$ and $s$ fixed, explicitly:\begin eqnarray*&&\int_01 \cdots \int_01 \left((r_1 - q_1)2 + \dots+(r_n-q_n)2\right)^ s/2 \ dr_1 \cdots dr_n.\end eqnarray* The study ofbox integrals leads one naturally into several disparate fields ofanalysis. While previous studies have focused upon symbolic evaluationand asymptotic analysis of special cases (notably $s = 1$), we workherein more generally--in interdisciplinary fashion--developing resultssuch as: (1) analytic continuation (in complex $s$), (2) relevantcombinatorial identities, (3) rapidly converging series, (4) statisticalinferences, (5) connections to mathematical physics, and ... continued below

## Who

People and organizations associated with either the creation of this article or its content.

### Provided By

#### UNT Libraries Government Documents Department

Serving as both a federal and a state depository library, the UNT Libraries Government Documents Department maintains millions of items in a variety of formats. The department is a member of the FDLP Content Partnerships Program and an Affiliated Archive of the National Archives.

## What

### Description

By a "box integral" we mean here an expectation $\langle|\vec r - \vec q|^s \rangle$ where $\vec r$runs over the unit $n$-cube,with $\vec q$ and $s$ fixed, explicitly:\begin eqnarray*&&\int_01 \cdots \int_01 \left((r_1 - q_1)2 + \dots+(r_n-q_n)2\right)^ s/2 \ dr_1 \cdots dr_n.\end eqnarray* The study ofbox integrals leads one naturally into several disparate fields ofanalysis. While previous studies have focused upon symbolic evaluationand asymptotic analysis of special cases (notably $s = 1$), we workherein more generally--in interdisciplinary fashion--developing resultssuch as: (1) analytic continuation (in complex $s$), (2) relevantcombinatorial identities, (3) rapidly converging series, (4) statisticalinferences, (5) connections to mathematical physics, and (6)extreme-precision quadrature techniques appropriate for these integrals.These intuitions and results open up avenues of experimental mathematics,with a view to new conjectures and theorems on integrals of thistype.

### Source

• Journal Name: Journal of Computational and AppliedMathematics; Journal Volume: 206; Journal Issue: 1; Related Information: Journal Publication Date: 09/2007

### Identifier

Unique identifying numbers for this article in the Digital Library or other systems.

• Report No.: LBNL--59962
• Grant Number: DE-AC02-05CH11231
• Office of Scientific & Technical Information Report Number: 919496

## When

• June 1, 2006

### Added to The UNT Digital Library

• Sept. 22, 2016, 2:13 a.m.

### Description Last Updated

• Sept. 30, 2016, 12:58 p.m.

Yesterday: 0
Past 30 days: 0
Total Uses: 1