{q^{\sigma+1} \over \sigma+1}
-\beta_{-1}(0) \log q +{\partial_s \beta_{-1}(0) \over N} \quad (q \to +\infty),$$ then on the definition (\[ZC\],\[DC\]), giving $$\label{SY}
{1 \over 2} \log D_{\rm cl}(\lambda) \defi  -{1 \over 2} \partial_s Z_{\rm cl}(s,\lambda)_{s=0}
= {\mathcal I}_0(\lambda) + 2(1-\log 2) {\beta_{-1}(0) \over N} .$$ The finite part ${\mathcal I}_0(\lambda)$ of $\int_0^{+\infty} (V(q)+\lambda) ^{-s+1/2} \d q$ at $s=0$ is a candidate to define a “symbolic" value for the divergent integral $\int_0^{+\infty} \Pi_\lambda(q) \d q$, where $$\label{PI}
\Pi_\lambda(q)  \defi (V(q)+\lambda) ^{1/2},  
\mbox{ the classical (forbidden-region) momentum} .$$ However, such an assignment being conventional, we much prefer the “renormalization" given by eq.(\[SY\]), which only adds an explicit constant to the finite part. So, (more generally) we pose the suggestive notation $$\label{INTP}
\int_q^{+\infty} \Pi_\lambda(q') \d q' \defi 
{1 \over 2} \log D_{\rm cl}(\lambda) - \int_0^q  \Pi_\lambda(q') \d q' 
\equiv {\mathcal I}_q(\lambda) + 2(1-\log 2) {\beta_{-1}(0) \over N} .$$ A big advantage of eq.(\[INTP\]) will be that its $\lambda \to +\infty$ expansion [*has the canonical form*]{} (\[das\]) (essentially because $\log D_{\rm cl}$ behaves similarly to its quantum counterpart $\log D$ in this respect, and $\int_0^q  \Pi_\lambda(q') \d q'=O(\sqrt \lambda)+o(1)$ is also manifestly canonical).

Thanks to eq.(\[I0\]), the prescription (\[INTP\]) is also directly characterized by its 