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I'm having some trouble deriving the expression $$\sigma_s(\omega) = \sigma_T \frac{(\frac{\omega_0}{2})^2}{(\omega - \omega_0)^2 + (\frac{\gamma}{2})^2}$$ from $$\sigma_s(\omega) = \sigma_T \frac{\omega^4}{(\omega^2 - \omega_0^2)^2 + \omega^2\gamma^2},$$

where $\omega - \omega_0 < \gamma$.

This is the resonant cross-section of electron scattering in electromagnetism and I can't find any derivations online, any help would be appreciated!

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The result is false in general: $\sigma_s(\omega=0)$ is finite for the first expression and zero for the second one, giving you a contradiction, and the condition $\omega-\omega_0<\gamma$ is satisfied at $\omega=0$ if $\gamma > 2\omega_0$, which is unreasonable but is consistent with the hypotheses you laid out.

The result you're probably looking for is an approximate equality between those two quantities that holds in the limit where $|\omega-\omega_0|\ll \gamma$, and that's a bit different. To do that, start with your initial expression, but factorize the denominator, and set $\omega=\omega_0+\delta$, giving you \begin{align} \sigma_s(\omega) & = \frac{\sigma_T \, \omega^4}{(\omega^2 - \omega_0^2)^2 + \omega^2\gamma^2} \\ & = \frac{\sigma_T \, \omega^4}{(\omega-\omega_0)^2(\omega+\omega_0)^2 + \omega^2\gamma^2} \\ & = \frac{\sigma_T \, (\omega_0+\delta)^4}{\delta^2(2\omega_0+\delta)^2 + (\omega_0+\delta)^2\gamma^2} \\ & \approx \frac{\sigma_T \, \omega_0^4}{4\delta^2\omega_0^2 + \omega_0^2\gamma^2} \\ & = \frac{\sigma_T \, \omega_0^2}{4\delta^2 + \gamma^2}, \end{align} which is the result you needed. The approximation as applied is setting $\omega\approx\omega_0$ except in those factors that contain the difference $\omega-\omega_0$, which is where the detuning really matters.

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