On a family of curious integrals suggested by Stellar Dynamics

TL;DR

This paper discovers and proves a new integral identity involving elliptic integrals, revealing a surprising simplification for a special case relevant to Stellar Dynamics, which was previously unknown to standard integral tables and software.

Contribution

The paper introduces a novel integral identity involving elliptic integrals, derived using elementary methods, and highlights its special case relevant to galaxy models in Stellar Dynamics.

Findings

Identified a new integral identity involving elliptic integrals.

Proved the identity using elementary methods.

Revealed a special case with a simple symbolic value.

Abstract

While investigating the properties of a galaxy model used in Stellar Dynamics, a curious integral identity was discovered. For a special value of a parameter, the identity reduces to a definite integral with a very simple symbolic value; but, quite surprisingly, all the consulted tables of integrals, and computer algebra systems, do not seem aware of this result. Here I show that this result is a special case ($n=0$ and $z=1$) of the following identity (established by elementary methods): $$ I_n(z)\equiv\int_0^1{{\rm K}(k) k\over (z+k^2)^{n+3/2}}dk = {(-2)^n\over (2n+1)!!} {d^n\over dz^n} {{\rm ArcCot}\sqrt{z}\over\sqrt{z(z+1)}},\quad z>0,$$ where $n=0,1,2,3...$, and ${\rm K}(k)$ is the complete elliptic integral of first kind.