The special case I$_3$ of the Kholodenko-Silagadze multiple integral considered anew

TL;DR

This paper revisits a complex multiple integral related to the Kholodenko-Silagadze quadrature, providing a new uniform quadrature reduction approach and resolving a previously unknown integral involving nd os functions.

Contribution

It introduces a novel uniform quadrature reduction perspective to analyze the integral and solves an unknown integral involving sine and cosine functions.

Findings

Confirmed the closed-form expression for the integral I_n for all n

Resolved a previously unknown integral involving sine and cosine functions

Demonstrated the effectiveness of the uniform quadrature reduction approach

Abstract

The nested Kholodenko-Silagadze quadrature \[ I_{n} = \int_{-\infty}^{\;\infty}ds_{1}\int_{-\infty}^{\;s_{1}}ds_{2}\int_{-\infty}^{\;s_{2}}ds_{3}\cdots \int_{-\infty}^{\;s_{2n-3}}ds_{2n-2}\int_{-\infty}^{\;s_{2n-2}}ds_{2n-1}\int_{-\infty}^{\;s_{2n-1}}ds_{2n}\cos(s_{1}^{2}-s_{2}^{2})\cos(s_{3}^{2}-s_{4}^{2})\cdots\cos(s_{2n-3}^{2}-s_{2n-2}^{2})\cos(s_{2n-1}^{2}-s_{2n}^{2})= \frac{2}{n!}\left(\frac{\pi}{4}\right)^{n} \;, \] obtained for all integers $n\geq 1$ by an elegant but indirect argument, is tackled anew from a uniform quadrature reduction viewpoint. Along the way, at its first instance of real difficulty when $n=3,$ the recondite quadrature \[ \int_{\,0}^{\;\infty} \frac{\cos(u)}{u} du \int_{\,0}^{\,u} \frac{\sin^{2}(v)}{v}dv + \int_{\,0}^{\;\infty} \frac{\sin(u)}{u} du \int_{\,0}^{\,u} \frac{\sin(v)\cos(v)}{v}dv = \,\frac{\pi^{2}}{12}\;,\] heretofore presumably unknown, receives an indirect resolution with its indicated value of $\pi^{2}/12$.