The Mellin transform and Euler-Maclaurin summation rule applied to system of n - independent harmonic oscillators

Abstract

We confront the Mellin and the Euler-Maclaurin summation rule when used in order to calculate N independent harmonic oscillators thermodynamic potentials, and compare the approximate expressions with the exact ones for these quantities. The goal of this work is at least twofold: First, to compare the results obtained for the thermodynamic potentials of a system of N oscillators using the Mellin and the Euler-Maclaurin summation rule in order to have a discernment on which one is most appropriate for application to more complex systems with finite and infinite number of components. Second, to present to the reader the ideas of these techniques in a pedestrian way with the aim of providing one material with detailed calculation that can be useful like a basic reference to more deep calculation, in particular string theory and supergravity.

Keywords

• partition function. harmonic oscillators
• thermodynamic potentials
• euler-maclaurin formula
• mellin transform.