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New formula for Asymptotic behavior of the Synchrotron function

Ivan Gonzalez, Daniel Salinas-Arizmendi

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.

New formula for Asymptotic behavior of the Synchrotron function

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.
Paper Structure (9 sections, 2 theorems, 60 equations, 2 figures, 2 tables)

This paper contains 9 sections, 2 theorems, 60 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The Method of Brackets
  3. Brackets series representations of Bessel function
  4. Asymptotic Expansion of $K_{\nu}(x)$ for large $x$
  5. Analytical Solution of the Synchrotron Function
  6. Small-$x$ asymptotics of $F(x)$
  7. Large-$x$ asymptotics of $F(x)$
  8. Conclusions
  9. Complete solution for large-$x$ values

Key Result

Theorem 3.1

The Brackets series representations of $K_{\nu}(x)$ Bessel function is given by

Figures (2)

  • Figure 1: Form of the divergent representation of the modified Bessel function $K_\nu(x)$ for $\nu=5/3$, derived from the bracket series. This asymptotic representation demonstrates convergence with numerically calculated values.
  • Figure 2: Analytical (gray dots) and numerical (green line) results for the synchrotron function $F(x)$ over the entire range of $x$.

Theorems & Definitions (4)

  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.4