Following Hardy, a Ramanujan  mock $vartheta$-function is a function defined by a $q$-series convergent when $|q|<1$, for which we can calculate asymptotic formulae, when $q$ tends to a ``rational point'' $e^{2rpi i/s}$ of the unit circle, of the same degree of precision as those furnished for the ordinary $vartheta$-functions by the theory of linear transformation. In our talk, we try to explain how to get the asymptotic behavior of third order mock theta functions, using Mordell integrals and Lerch series.