A convergent version of Watson's lemma for double integrals

A modification of Watson's lemma for Laplace transforms f(0)(infinity) f(t) e(-zt) dt was introduced in [Nielsen, 1906], deriving a new asymptotic expansion for large vertical bar z vertical bar with the extra property of being convergent as well. Inspired in that idea, in this paper we derive asymptotic expansions of two-dimensional Laplace transforms F(x,y) := f(0)(infinity) f(0)(infinity )f(t, s) e(-xt-ys) dtds for large vertical bar x vertical bar and vertical bar y vertical bar that are also convergent. 0 The expansions of F(x,y) are accompanied by error bounds. Asymptotic and convergent expansions of some special functions are given as illustration.