A family of Exponential Integrals suggested by Stellar Dynamics

TL;DR

This paper derives a family of exponential integrals related to stellar dynamics, providing elementary methods to evaluate them and connecting them to special functions like the incomplete gamma function.

Contribution

It introduces a new family of exponential integrals and demonstrates how to evaluate them using elementary methods, expanding the understanding of integrals arising in stellar dynamics.

Findings

Derived a general formula for the family of integrals involving exponential integrals.

Connected the integrals to special functions such as the incomplete gamma function.

Provided elementary evaluation methods for these integrals.

Abstract

While investigating the generalization of the Chandrasekhar (1943) dynamical friction to the case of field stars with a power-law mass spectrum and equipartition Maxwell-Boltzmann velocity distribution, a pair of 2-dimensional integrals involving the Error function occurred, with closed form solution in terms of Exponential Integrals (Ciotti 2010). Here we show that both the integrals are very special cases of the family of (real) functions $$ I(\lambda,\mu,\nu; z) :=\int_0^zx^{\lambda}\,\Enu(x^{\mu})\,dx= {\gamma\left({1+\lambda\over\mu},z^{\mu}\right) + z^{1+\lambda}\Enu(z^{\mu})\over 1+\lambda + \mu (\nu -1)}, \quad \mu>0,\quad z\geq 0, \eqno (1) $$ where $\Enu$ is the Exponential Integral, $\gamma$ is the incomplete Euler gamma function, and for existence $\lambda >\max \left\{-1,-1- \mu(\nu -1)\right\}$. Only in one of the consulted tables a related integral appears, that with some work can be reduced to eq.~(1), while computer algebra systems seem to be able to evaluate the integral in closed (and more complicated) form only provided numerical values for some of the parameters are assigned. Here we show how eq.~(1) can in fact be established by elementary methods.