Synchrotron function

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

• First synchrotron function

$f(x)=x\int _{x}^{\infty }K_{\frac {5}{3}}(t)\,dt$

• Second synchrotron function

$f(x)=xK_{\frac {2}{3}}(x)$

where K_j is the modified Bessel function of the second kind. This is needed when calculating the spectra for different types of synchrotron emssion. It takes a number of spectrum of electrons generated by a seperate process (such as a power law distribution of electrons and positrons) and converts this to the spectrum of photons generated in the magnetic field by the input electrons.

For further information, see High Energy Astrophysics, Malcolm S. Longair, CUP, 1990.