A Laplace integral on a Kähler manifold and Calabi's diastasis function

In this paper we give a different proof of EngliÅ¡'s result [J. Reine Angew. Math. 528 (2000) 1-39] about the asymptotic expansion of a Laplace integral on a real analytic Kähler manifold (M,g) by using the link between the metric g and the associated Calabi's diastasis function D. We also make explicit the connection between the coefficients of EngliÅ¡' expansion and Gray's invariants [Michigan Math. J. (1973) 329-344].