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Soft Photon Limit

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 [Graphics:SoftComptongr55.gif]

Allowed range of &Dgr; and [Graphics:SoftComptongr70.gif] : changing the order of integration

Average of cross-section over [Graphics:SoftComptongr94.gif] for fixed &Dgr;

Weighted Average of cross-section over [Graphics:SoftComptongr126.gif] for fixed &Dgr;

Thermal average of moments of &Dgr; (i.e. moments as function of [Graphics:SoftComptongr216.gif])

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

First Order Correction for Harder Photons

Slow electron limit

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