Error-function logarithmic switchback ($-\frac{1}{4}\epsilon \log \epsilon \mathrm{erf}z$)

If an outer expansion contains $-(\epsilon /2)\log x$ and the layer coordinate is $x=\sqrt{\epsilon }z$, its overlap contains $-(\epsilon /2)\log z-(\epsilon /4)\log \epsilon$. Matching forces an inner expansion term $-(\epsilon /4)\log \epsilon \mathrm{erf}z$ when that homogeneous profile tends to one in the overlap and vanishes at the boundary. The logarithm multiplies the matching profile, rather than an arbitrary constant.