This is the generating function for all of the integrals we want.
Sometimes a simpler relation is obtained by using rerurrence relation to express a function in terms of Kn(U) and Kn-1(U) for n>1 instead of K1(U) and K2(U). We may do this using the
Phase space measure
First solve for &bgr; as a function of &ggr;
Of course the postive root is what we want!
Now compute d(&bgr;&ggr;)/d&ggr; which gives the dp in the momentum integral.
and finally the complete momentum phase space measure p2dp/d&ggr;
To normalize our thermal distribution we set integral of eE/kTp2dp to unity. Of course for an electron E=&ggr; and use use variable .
To do the integral we expand the integrand divided by Sqrt[&ggr;2-1] and e-&ggr;U and expand in powers of &ggr; (apart form these factors the integrands are polynomials in &ggr;) . The function ThermalIntegral[n,U] defined above gives the integral for &ggr;n.
Document converted by Mathematica of Wolfram Research