points of f are non-degenerate and isolated, then the question can be reduced to the case n = 1. In fact, then, a choice of g can be made to split the integral into cases with just one critical point P in each. At that point, because the Hessian determinant at P is by assumption not 0, the Morse lemma applies. By a change of co-ordinates f may be replaced by . The value of j is given by the signature of the Hessian matrix of f at P. As for g, the essential case is that g is a product of bump functions of xi. Assuming now without loss of generality that P is the origin, take a smooth bump function h with value 1 on the interval and quickly tending to 0 outside it. Take , then Fubini's theorem reduces I(k) to a product of integrals over the real line like with f(x) = ±x2. The case with the minus sign is the complex conjugate of the case with the plus sign, so there is essentially one required asymptotic estimate. In this way asymptotics can be found for oscillatory integrals for Morse functions. The degenerate case requires further techniques (see for example Airy function). One-dimensional case The essential statement is this one: . In fact by contour integration it can be shown that the main term on the right hand side of the equation is the value of the integral on the left hand side, extended over the range (for a proof see Fresnel integral). Therefore it is the question of estimating away the integral over, say, . This is the model for all one-dimensional integrals with having a single non-degenerate critical point at which has second derivative . In fact the model case has second derivative 2 at 0. In order to scale using , observe that replacing by where is constant is the same as scaling by . It follows that for general values of , the factor becomes . For one uses the complex conjugate formula, as mentioned before. Lower-order terms As can be seen from the formula, the stationary phase approximation is a first-order approximation of the asymptotic behavior of the integral. The lower-order terms can be understood as a sum of over Feynman diagrams with various weighting factors, for well behaved . See also Common integrals in quantum field theory Laplace's method Method