Stationary phase method

Oscillatory integrals are led by points where the phase derivative vanishes, with a phase shift determined by the Hessian signature.

If the first two phase derivatives vanish at an endpoint but the third does not, an oscillatory contribution is concentrated on a width proportional to $N^{-1/3}$. For a smooth nonzero endpoint oscillatory integral amplitude, rescaling reduces the leading contribution to a cubic oscillatory integral. Its constant follows from the Gamma function Fourier integral. This is a degenerate form of the stationary phase method, and a quadratic formula is insufficient.

At equal argument and large integer order, the Bessel function of the first kind has phase $n(\sin t-t)=-n{t}^3/6+\cdots$ near the endpoint. The cubic stationary endpoint yields the displayed positive constant. It is also the zero-argument value of the Airy function in the uniform Bessel turning-point approximation.

In an integral $\int a(q)e^{i\lambda \beta (q)}dq$, the real-valued function $\beta$ is the phase. Its stationary points satisfy $\nabla \beta =0$; its Hessian matrix determines the local quadratic oscillation and the signature factor in stationary phase method. Degenerate stationary points and boundaries require separate analysis.

For a smooth real phase with an isolated nondegenerate interior critical point $q$, smooth amplitude $a$, and no competing boundary contribution, the leading contribution is $2\pi {\lambda }^{-1}a(q)|\det H{|}^{-1/2}{e}^{i\lambda \varphi (q)+i\pi \mathrm{sgn}(H)/4}$, where $H$ is the Hessian matrix. Orthogonal diagonalization gives two one-dimensional Fresnel integrals. A degenerate critical point or a stationary boundary needs its own local analysis.

For fixed $\nu$, the vertical sides of the half-strip integral of $e^{-ix\sin z+i\nu z}$ contribute $-\sin (\pi \nu )/(\pi x)+O(x^{-2})$. Integration by parts on the nonstationary horizontal pieces contributes the opposite endpoint term. After cancellation, the two stationary points give $\sqrt{2/(\pi x)}\cos (x-\nu \pi /2-\pi /4)+O(x^{-3/2})$.

A stationary point $x_0$ of a smooth phase $\varphi$ is nondegenerate when its Hessian matrix at $x_0$ is an invertible matrix. In one dimension, this condition is ${\varphi }^{\prime \prime }(x_0)\ne 0$.

If ${\psi }^{\prime }(k_0)=0$, ${\psi }^{\prime \prime }(k_0)\ne 0$, and $A$ is smooth near $k_0$, thenContributions from distinct stationary points add.

If a phase at an endpoint satisfies $\varphi (u)=\varphi (0)-c{u}^3+O(u^5)$ with $c>0$, the leading region has width $O(t^{-1/3})$. Rescaling $u=(ct{)}^{-1/3}v$ reduces the leading term to a cubic oscillatory integral and produces decay $t^{-1/3}$.

Forfixed $\alpha$ and $x\to +\infty$, the stationary point near $\pi /2$ givesWhen $\alpha =x$, the stationary point coalesces cubically with the endpoint at zero and the leading order changes to $x^{-1/3}$.

If the phase derivative has no zero on the support of a smooth amplitude, repeated integration by parts makes the oscillatory integral decay faster than any prescribed inverse power, subject to corresponding decay and regularity of the amplitude.