In mathematics the synchrotron functions are defined as follows (for x ≥ 0):
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 Kj is the modified Bessel function of the second kind.
First and second Synchrotron functions.
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.
For further information, see High Energy Astrophysics, Malcolm S. Longair, CUP, 1990.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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