Auxillary Functions

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

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Namely [Graphics:SoftComptongr130.gif]==[Graphics:SoftComptongr131.gif] [Graphics:SoftComptongr132.gif][Graphics:SoftComptongr133.gif]

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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
function

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Phase space measure

First solve for &bgr; as a function of &ggr;

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Of course the postive root is what we want!

Now compute d(&bgr;&ggr;)/d&ggr; which gives the dp in the momentum integral.

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and finally the complete momentum phase space measure p2dp/d&ggr;

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Normalizing Integral

To normalize our thermal distribution we set integral of  eE/kTp2dp to unity. Of course for an electron E=&ggr;[Graphics:SoftComptongr148.gif][Graphics:SoftComptongr149.gif] and use use variable [Graphics:SoftComptongr150.gif].

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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