In mathematics the synchrotron functions are defined as follows (for x ≥ 0):^[1]

• First synchrotron function $${\displaystyle F(x)=x\int _{x}^{\infty }K_{\frac {5}{3}}(t)\,dt}$$
• Second synchrotron function $${\displaystyle G(x)=xK_{\frac {2}{3}}(x)}$$

where K_j is the modified Bessel function of the second kind.

In astrophysics, x is usually a ratio of frequencies, that is, the frequency over a critical frequency (critical frequency is the frequency at which most synchrotron radiation is radiated). This is needed when calculating the spectra for different types of synchrotron emission. It takes a spectrum of electrons (or any charged particle) generated by a separate process (such as a power law distribution of electrons and positrons from a constant injection spectrum) and converts this to the spectrum of photons generated by the input electrons/positrons.

• Longair, Malcolm S. (2011). High energy astrophysics (3rd ed.). Cambridge: Cambridge University Press. ISBN 978-0-511-93059-1. OCLC 702125055.
• Rybicki, George B. (2004). Radiative processes in astrophysics (PDF). Alan P. Lightman. Weinheim. p. 191. ISBN 978-3-527-61817-0. OCLC 212140606.{{cite book}}: CS1 maint: location missing publisher (link)