  Let $q=e^{2\pi i\tau}$, $\Im\tau>0$, $x=e^{2\pi i\xi}\in\CC$ and
$(x;q)_\infty=\prod_{n\ge 0}(1-xq^n)$. Let $(q,x)\mapsto(q^*,\iota_q x)$ be the
classical modular substitution given by $q^*=e^{-2\pi i/\tau}$ and $\iota_q
x=e^{2\pi i\xi/{\tau}}$. The main goal of this Note is to study the "modular
behaviour" of the infinite product $(x;q)_\infty$, this means, to compare the
function defined by $(x;q)_\infty$ with that given by $(\iota_q x;q^*)_\infty$.
Inspired by the work of Stieltjes on some semi-convergent series, we are led to
a "closed" analytic formula for $(x;q)_\infty$ by means of the dilogarithm
combined with a Laplace type integral that admits a divergent series as Taylor
expansion at $\log q=0$. Thus, we can obtain an expression linking
$(x;q)_\infty$ to its modular transform $(\iota_qx;q^*)_\infty$ and which
contains, in essence, the modular formulae known for Dedekind's eta function,
Jacobi theta function and also for certain Lambert series. Among other
applications, one can remark that our results allow to obtain a Ramanujan's
asymptotic formula about $(x;q)_\infty$ for $q\to 1$.
