A convergent version of Watson's lemma for double integrals

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