= Neutral-mode critical-layer matching
{title2=$b_+-b_-=i\pi c_1U'''(0)\phi_0(0)/U'(0)^2$}
For a monotone <inviscid parallel shear flow>, let a regular neutral <Rayleigh equation for inviscid shear flow> mode have $U(0)=c_0$, $U''(0)=0$, $d=U'(0)>0$, and $p=\phi_0(0)\ne0$. A perturbation $c=c_0+\varepsilon c_1$ with $\operatorname{Im}c_1>0$ resolves the singularity on the <inner variable> $\eta=y/\varepsilon$. The first outer correction has $\phi_1=a+\beta y\log|y|+b_\pm y+\cdots$, with $\beta=c_1U'''(0)p/d^2$. The inner particular solution contains $(d\eta-c_1)\log(d\eta-c_1)$. Its <complex logarithm> has argument tending to zero at positive infinity and to $-\pi$ at negative infinity, giving the displayed matching jump.
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