This is the generating function for all of the integrals we want.

Namely **==**** **

Sometimes a simpler relation is obtained by using rerurrence relation to express a function in terms of *K**n*(*U*) and *K*n-1(*U*) for *n*>1 instead of *K*1(*U*) and *K*2(*U*). We may do this using the

function

*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 *p*2*dp*/*d*&ggr;

*Normalizing Integral*

To normalize our thermal distribution we set integral of e*E*/*kT**p*2*dp* 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.

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