New formula for Asymptotic behavior of the Synchrotron function

TL;DR

This paper introduces a new analytical formula for the synchrotron function, enabling accurate asymptotic approximations and offering a faster alternative to numerical methods for astrophysical applications.

Contribution

It presents a novel compact analytical representation of the synchrotron function using the Method of Brackets, improving efficiency and understanding of its asymptotic behavior.

Findings

Accurately reproduces numerical integration results.

Provides explicit analytic structure of the synchrotron function.

Offers efficient asymptotic expansions for small and large arguments.

Abstract

Synchrotron radiation plays a central role in astrophysical and high-energy processes. Its spectral description involves the synchrotron function, defined by a non-trivial integral of modified Bessel functions and commonly evaluated through numerical methods or dedicated approximations. In this work, we obtain a compact analytical representation of the synchrotron function using the Method of Brackets, which yields systematically controllable asymptotic expansions in both the small- and large-argument regimes. The resulting expressions accurately reproduce numerical integration and make the analytic structure of the function explicit. Our results provide an efficient alternative to repeated numerical evaluations and facilitate applications requiring fast and controlled approximations.