= Damped first-order resonant complex eccentricity
{title2=$\dot z=-iC-(A+iBx)z$}
For an exterior $(p+1):p$ <mean-motion resonance>, use <mean longitude> in $\phi=(p+1)\lambda-p\lambda_p-\varpi$, $z=e\exp(i\phi)$, $x=(a-a_r)/a_r$ and $B=3pn_p/2$. Given $\dot e=-C\sin\phi-Ae$ and $\dot\varpi=(C/e)\cos\phi$, the leading Keplerian detuning gives $\dot\phi=-Bx-(C/e)\cos\phi$. Differentiating $z$ and using $e^{i\phi}(\sin\phi+i\cos\phi)=i$ proves the displayed equation. It retains resonant forcing and <eccentricity damping> without the coordinate singularity at $e=0$. True longitude adds a fast orbital ripple and does not yield this autonomous averaged equation directly.
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