Analytical Fits to the Synchrotron Functions

Abstract

Accurate fitting formulae to the synchrotron function, $F(x)$, and its complementary function, $G(x)$, are performed and presented. The corresponding relative errors are less than $0.26\%$ and $0.035\%$ for $F(x) $ and $G(x)$, respectively. To this aim we have, first, fitted the modified Bessel functions, $K_{5/3}(x)$ and $K_{2/3}(x)$. For all the fitted functions, the general fit expression is the same, and is based on the well known asymptotic forms for low and large $x$-values for each function. It consists of multiplying each asymptotic form by a function that tends to unity or zero for low and large $x$-values. Simple formulae are suggested in this paper, depending on adjustable parameters. The latter have been determined by adopting the Levenberg-Marquardt algorithm. The proposed formulae should be of great utility and simplicity for computing spectral powers and the degree of polarization for the synchrotron radiation, both for laboratory and astrophysical applications.