thumb|First and second Synchrotron functions.
In mathematics the synchrotron functions are defined as follows (for x ≥ 0):
- First synchrotron function <math display="block">F(x) = x \int_x^\infty K_{\frac{5}{3(t)\,dt</math>
- Second synchrotron function <math display="block">G(x) = x K_{\frac{2}{3(x)</math>
where K<sub>j</sub> is the modified Bessel function of the second kind.
Use in astrophysics
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.