This note deals with integrals of the form
where (the Schwartz space), , , and with , , 1, . The asymptotic expansion of I(s,f) as is derived by induction on n using a two-variable asymptotic expansion in the induction step. As a special case one obtains a recent result of D. Barlet [Invent. Math. 68, 129-174 (1982; Zbl 0508.32003)].