Lorentzian resonance approximation ($\gamma /(x^2+{\gamma }^2)\to \pi \delta (x)$)

For $\gamma >0$, the kernels $\gamma /[\pi (x^2+{\gamma }^2)]$ have unit integral and concentrate at zero as $\gamma \to 0^{+}$. Splitting the integral into a neighborhood of zero and its complement proves convergence against bounded continuous test functions. Thus $[(kv-\omega {)}^2+{\gamma }^2{]}^{-1}\simeq \pi \delta (kv-\omega )/|\gamma |$ under an integral, with $\delta (kv-\omega )=\delta (v-\omega /k)/|k|$. This is the narrow-width limit of a Cauchy distribution, not a pointwise replacement away from the peak.