= Lorentzian resonance approximation
{c}
{title2=$\gamma/(x^2+\gamma^2)\to\pi\delta(x)$}
For $\gamma>0$, the kernels $\gamma/[\pi(x^2+\gamma^2)]$ have unit integral and concentrate at zero as $\gamma\to0^+$. 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.
Back to article page
