Endpoint transition of a regularized interior pole ($I\sim [\pi /2+\arctan (d/(\eta L_0))]/(a\eta L_0)$)

Suppose $a,b,L_0>0$, $L(x)$ is smooth and positive near zero with $L(0)=L_0$, and $d=O(\eta )$ as $\eta \to 0^{+}$. The leading contribution tois obtained by scaling $x=\eta s$ and replacing $L$ by $L_0$. Integration of the resulting reciprocal quadratic gives the displayed formula. For constant $L$ the exact expression isThis proves the half-peak value at $d=0$ and the transition between an interior peak and a pole outside the interval. If $L$ varies, smoothness and positivity on the contributing neighborhood justify its leading replacement; other zeros or singular contributions elsewhere must be excluded.

If $h$ has a simple interior zero $u_0$ and $a\to 0^{+}$, the inner variable $v=h^{\prime }(u_0)(u-u_0)/a$ turns the local integral into $(1/|h^{\prime }(u_0)|){\int }_{\mathbb{R}}(1+v^2{)}^{-1}dv$. 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.