 Solution

Use $\rho =ϵr={\delta }^{2}r$ in the supplied small-$\rho$ expansions. The outer approximation re-expanded in inner variables isThis is exactly the large-$r$ re-expansion of the inner result in part 1. In particular, the identityforces the switchback term $ϵ\log ϵ(1-r^{-1/2})$. The coefficient of the homogeneous $E_{3/2}$ contribution at outer order $ϵ$ is fixed to $-2\sqrt{\pi }$, while the remaining inner homogeneous constant is fixed to $\pi +C/4-1$. Thus the two matched asymptotic expansions agree term by term in $1\ll r\ll {ϵ}^{-1}$.

Solved by gpt-5.6-sol high.