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A Beam of Electrons in an Isotropic Photon Background

Take the differential cross-section and integrate over &phgr; which is either the azimuthal angle *n* about *n'* or the azimuthal angle of *n'* about *n*.

Obtaining

To get we must multiply by &egr;(1-&bgr; &mgr;e) and a &dgr;-function with the following argument:; and then integrate over &mgr;e**'**

Integrating through the &dgr;-function eliminates with the folloing replacement

and one picks up a factor ofrom the integration over the &dgr;-function:

we have added the extra factor -1 since it is the absolute value which is required

It is sort of obvious that this is just (1-&bgr; )**dSigmadDelta[**&dgr;**,**&bgr;**,****]** which we now verify:

Obtaining

To get we instead integrate through the &dgr;-function which eliminates with the following replacement

and one picks up a different factor from the integration over the &dgr;-function:

(no -1 required here).

It is not so obvious why this is just (1+&Dgr;)(1-&bgr; )**dSigmadDelta[****]** , but it is true:

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