# 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

### Collections

#### Office of Scientific & Technical Information Technical Reports

Reports, articles and other documents harvested from the Office of Scientific and Technical Information.

Office of Scientific and Technical Information (OSTI) is the Department of Energy (DOE) office that collects, preserves, and disseminates DOE-sponsored research and development (R&D) results that are the outcomes of R&D projects or other funded activities at DOE labs and facilities nationwide and grantees at universities and other institutions.

## 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: 2