Show that for light with a Lorentzian spectral profile, the parameter (mathcal{M}) is given by [ mathcal{M}=left[frac{tau_{c}}{T}+frac{1}{2}left(frac{tau_{c}}{T} ight)^{2}left(e^{-2 T / tau_{c}}-1 ight) ight]^{-1} ]

Show that for light with a Lorentzian spectral profile, the parameter \(\mathcal{M}\) is given by

\[ \mathcal{M}=\left[\frac{\tau_{c}}{T}+\frac{1}{2}\left(\frac{\tau_{c}}{T}\right)^{2}\left(e^{-2 T / \tau_{c}}-1\right)\right]^{-1} \]