  In this paper we employ a novel technique combining the Euler Maclaurin
formula with the saddle point approximation method to obtain the asymptotic
behavior (in the limit of large representation index $J$) of generic Wigner
matrix elements $D^{J}_{MM'}(g)$. We use this result to derive asymptotic
formulae for the character $\chi^J(g)$ of an SU(2) group element and for
Wigner's $3j$ symbol. Surprisingly, given that we perform five successive
layers of approximations, the asymptotic formula we obtain for $\chi^J(g)$ is
in fact exact. This result provides a non trivial example of a
Duistermaat-Heckman like localization property for discrete sums.
