Lemma: Exact local form at a strictly diffracted front

Near a point with $F_{\mathrm{ap}}=0$ and $\delta \notin {\mathcal{B}}_{\alpha }$, there are smooth functions $A_{\ast }$ and $B_{\ast }$, analytic in the local real variables, such that

and

The square root is the principal boundary value.

Proof. Take first $T>0$. Near $Q=-1$, write $\eta =\mathrm{i}\pi +w$. Since

one has

The angular denominator in (eq:Sclosed) is nonzero at $w=0$, and

Expand $S_{\beta }(\mathrm{i}\pi +w,\delta )/\sinh (\mathrm{i}\pi +w)$ in its convergent Laurent series in $w$. Its even and odd powers separate into an analytic function of $F_{\mathrm{ap}}$ plus $(F_{\mathrm{ap}}+\mathrm{i}0{)}^{-1/2}$ times another analytic function. Multiplication by the prefactor in (eq:Wclosed) gives (eq:diffcoefficient).

For $T<0$, expand at $\eta =-\mathrm{i}\pi$. Oddness of $S_{\beta }$ and of $\sinh \eta$ gives the same coefficient and the lower boundary value.