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Differential cross-section wrt &thgr; and  &phgr;

Change of variables from &thgr; to &Dgr;

Differential cross-section wrt &Dgr;

Moments of &Dgr; as function of &bgr; and

Allowed range of &Dgr; and : changing the order of integration

Average of cross-section over for fixed &Dgr;

Weighted Average of cross-section over for fixed &Dgr;

Thermal average of moments of &Dgr; (i.e. moments as function of )

Thermal average expansion of moments in small temperature

Extended Kompaneets Equation

Extended Sunyaev Zel'dovich Distortion

Comparison With Challinor and Lasenby

Comparison With Nozawa, Itoh, and Kohyama (astro-ph/9804051)

Non-Perturbative Rayleigh-Jeans Limit

Thermal Velocity Effect

Finite Optical Depth Effects

Plot of Rayeigh-Jeans Temperature for Fiducial Cluster

PLANCK's sensitivity to Extended S-Z effect

A Beam of Electrons in an Isotropic Photon Background

Slow electron limit

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