Laplace's method and steepest descent

For a clear real-variable theorem, take a smooth compactly supported amplitude on the real line and a real smooth phase near its support. Assume the phase has a unique global minimum on that support at an interior point where the amplitude is nonzero, its first derivative vanishes, and its second derivative is strictly positive. Compactness and uniqueness separate the phase from its minimum outside any small neighborhood. Thus the distant contribution is exponentially suppressed, while the nearby quadratic expansion determines the leading term.

Set the distance from the minimum equal to a new variable divided by the square root of the large parameter. The quadratic phase becomes order one and the integration measure contributes the inverse square root. Smooth expansion of the amplitude and higher phase terms then produces corrections. Under the stated smooth compact-support assumptions, the leading Gaussian term has an absolute remainder of order the parameter to minus three-halves times the exponential minimum factor. Boundary minima or degenerate critical points require different powers and constants, so check the geometry before using the formula.

For a quadratic phase and a polynomial amplitude, some of the asymptotic calculation can be exact. The integral of a Gaussian is the square root of pi divided by the parameter. Differentiating that identity with respect to a positive parameter, justified by Gaussian domination on compact positive parameter ranges, gives its second-moment integral. Adding those two contributions evaluates an amplitude equal to one plus the squared coordinate. This is an improper-integral example with explicit tail control rather than an unproved extension of the compact-support theorem.

The correction is smaller than the leading term by the reciprocal of twice the large parameter. Relative error must specify its denominator: the omitted correction divided by the leading approximation differs from the omitted correction divided by the exact positive integral. Exact examples are valuable for checking these distinctions and normalizing constants. They do not establish that a leading-term approximation has the same finite-parameter error for every amplitude or phase with a similar-looking minimum.

For a holomorphic amplitude and phase, a steepest-descent contour through a nondegenerate saddle keeps the phase's imaginary part constant locally and makes its real part increase away from the saddle for an integrand with negative exponential phase. Cauchy's theorem permits a deformation only through a region where the integrand is holomorphic, with endpoint and infinite-tail contributions controlled. Crossing a pole adds residue information; crossing a branch cut changes the continuation problem. Locating a saddle alone does not establish that its steepest path belongs to the original integral.

The orientation of the local contour fixes the phase of the Gaussian square root. Several accessible saddles may contribute, and their relative importance can change as parameters move. A coalescing saddle or a saddle approaching an endpoint needs a uniform canonical approximation rather than separate nondegenerate formulas with exploding coefficients. The contour-shift practice below is deliberately entire and quadratic, so the connecting vertical edges can be bounded explicitly. That gives a complete deformation argument without concealing singularities or unverified tail behavior.