= Lorentzian contribution from shrinking regularization
{c}
{title2=$\int a\,[h(u)^2+a^2]^{-1}du\to\pi/|h'(u_0)|$}
If $h$ has a simple interior zero $u_0$ and $a\to0^+$, the <inner variable> $v=h'(u_0)(u-u_0)/a$ turns the local integral into $(1/|h'(u_0)|)\int_{\mathbb R}(1+v^2)^{-1}dv$. Thus a peak with vanishing width can retain finite area. Slowly varying positive regularization factors cancel between peak height and width when the numerator has the matching factor. An ordinary <outer expansion> can miss this contribution even though the peak is exponentially narrow.
Back to article page
