Recently Naoto Kumano-go (Bulletin des Sciences Mathématiques,vol.128 (2004)) succeeded in proving that piecewise linear time slicing approximation to Feynman path integral with integrand F (γ) actually converges to the limit if the functional F (γ) of paths γ belongs to a certain class of functionals, which includes, as a typical example, Stieltjes integral of the following form;

F (γ) = ∫_{0}^{T} f(t, γ(t))ρ(dt),  (1)

here ρ(t) is a function of bounded variation and f(t, x) is a sufficiently smooth function with polynomial growth at |x| → ∞.

We shall write down the second term of the semi-classical asymptotic series of the Feynman path integrals if the integrand is of the form (1). If F (γ) ≡ 1, this second term coincides with the one given by G.D. Birkhoff in his famous paper ( Bull. Amer. Math. Soc. vol 39,(1933 ). Otherwise, we believe the formula is new as far as we know.