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

Asymptotes in SU(2) Recoupling Theory: Wigner Matrices, $3j$ Symbols, and Character Localization