<s>[INST] Can this approximation be made more formal? When considering oscillating systems in physics, we end up with some response function like $$F(\omega) = \frac{\omega^2}{(\omega_0^2 - \omega^2)^2 + (\omega/\tau)^2},$$ where $\omega_0$ and $\tau$ are characteristic properties of the system, and $\omega$ is the driving frequency. We are generally interested in the behavior of $F(\omega)$ close to the maximum, $\omega = \omega_0$, which is the resonant frequency. However, the usual approach to approximate $F(\omega)$ is the following $$F(\omega) = \frac{\omega^2}{(\omega_0 - \omega)^2 (\omega_0 + \omega)^2 + (\omega/\tau)^2} \approx \frac{\omega_0^2}{4 \omega_0^2 (\omega_0 - \omega)^2 + (\omega_0/\tau)^2} = \frac{\frac{1}{4}}{(\omega_0 - \omega)^2 + (1/2\tau)^2}.$$
Is there a way to formalize and justify the above approximation? Could this be a form of Pade approximation?
 [/INST] Let $F(\omega)$ be given by
$$\begin{align}
F(\omega)&=\frac{\omega^2}{(\omega^2-\omega_0^2)^2+(\omega/\tau)^2}\\\\
&=\frac{1/4}{(\omega_0/2)^2(1-\omega/\omega_0)^2(1+\omega/\omega_0)^2+(1/2\tau)^2}
\end{align}$$
Now, let's denote $\omega/\omega_0=1+x$.  Then, we have
$$\begin{align}
F(\omega)&=F(\omega_0(1+x))\\\\
&=\frac{1/4}{(\omega_0x)^2\,(1+x/2)^2+(1/2\tau)^2}\\\\
&=\frac{1/4}{(\omega_0x)^2+(1/2\tau)^2}\times\left(\frac{(\omega_0x)^2+(1/2\tau)^2}{(\omega_0x)^2\,(1+x/2)^2+(1/2\tau)^2}\right)\\\\
&=\frac{1/4}{(\omega_0x)^2+(1/2\tau)^2}\times\left(1+O\left(x^3\right)\right)\\\\
&=\frac{1/4}{(\omega-\omega_0)^2+(1/2\tau)^2}\times\left(1+O\left((\omega-\omega_0)^3\right)\right)
\end{align}$$
</s>