\infty,$$ even though $D_\infty^\pm(0)$ have finite values, perfectly defined by eq.(\[Di0\]), in the $N=+\infty$ problem ! (All that shows how carefully such transitional problems must be handled. — The same formulae show the [*skew*]{} spectral functions (\[SK\]) to behave slightly better: e.g., $Z_N^{\rm P}(0) \equiv 1/2 = Z_\infty^{\rm P}(0)$, $D_N^{\rm P}(0) \to 1 = D_\infty^{\rm P}(0)$.)

We then basically expect $$\label{Dreg}
\Delta_N^\pm (\lambda) \to \Delta_\infty^\pm (\lambda) \quad
\mbox{ but }D_N^\pm(\lambda) \mbox{ \sl diverge\/},$$ this divergence being confined here to the factor $\exp [- Z'_N(0)] = D_N(0)$ alone, because $\mu_\infty<1$ and the Fredholm determinants can be expressed using $s=1,2,\cdots$ only as in eq.(\[DFr\]). (The Fredholm determinants $\Delta(\lambda)$ should behave regularly [*in general*]{} since they are designed by retaining only [*regular*]{} values $Z(s)$ (at integers $s$) in their Taylor series (cf. eq.(\[Dtay\])); this has to be qualified only if $\mu$ reaches (or jumps across) an integer in the limit, as in the example of Sec.3.)

The main functional relation
----------------------------

We now study the $N \to +\infty$ limit of the functional relation (\[DW\]) specialized to homogeneous potentials $|q|^N$, as stated in eq.(\[bfr\]). By the preceding arguments, this functional relation should be well-behaved as $N \to +\infty$ only once it has been transcribed 