Neutral-mode critical-layer matching ($b_{+}-b_{-}=i\pi c_1{U}^{\prime \prime \prime }(0){\varphi }_0(0)/U^{\prime }(0{)}^2$)

For a monotone inviscid parallel shear flow, let a regular neutral Rayleigh equation for inviscid shear flow mode have $U(0)=c_0$, $U^{\prime \prime }(0)=0$, $d=U^{\prime }(0)>0$, and $p={\varphi }_0(0)\ne 0$. A perturbation $c=c_0+\epsilon c_1$ with $\mathrm{Im}{c}_1>0$ resolves the singularity on the inner variable $\eta =y/\epsilon$. The first outer correction has ${\varphi }_1=a+\beta y\log |y|+b_{\pm }y+\cdots$, with $\beta =c_1{U}^{\prime \prime \prime }(0)p/d^2$. 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.

Let $I=\int {\varphi }_0^{2}dy$, $J=\mathrm{PV}\int U^{\prime \prime }{\varphi }_0^2/(U-c_0{)}^{2}dy$, and $M=U^{\prime \prime \prime }(0){\varphi }_0(0{)}^2/U^{\prime }(0{)}^2$. The Wronskian identity for the first-order Rayleigh equation for inviscid shear flow, combined with neutral-mode critical-layer matching, gives $2k_0{k}_{1}I=-c_1(J+i\pi M)$. For real $k_1$, its imaginary part is $2k_0{k}_{1}I\pi M/(J^2+{\pi }^2{M}^2)$. The Cauchy principal value is necessary when the outer forcing has a simple real pole.