Appendix A

The stationary phase theorem announces the following result Bleistein (1984):

Integrals of the form

$$\begin{eqnarray} I(k) = \int_{-\infty}^{+\infty} e^{ik\phi(t)} f(t) dt\end{eqnarray}$$ (12)

can be approximated asymptotically when $k \rightarrow \infty$ by:

$$\begin{eqnarray} I(k) \approx \sqrt{ \frac{2\pi}{k\vert\phi''(t_0)\vert} } f(t_0) e^{ik\phi(t_0) + sgn(\phi''(t_0)) \frac{i\pi}{4}}\end{eqnarray}$$ (13)

where t₀ is the stationary point at which the phase derivative $\phi'(t)=0$, f(t) is a complex function, and the phase $\phi(t)$ is real.

The method is based on a high-frequency approximation, the general idea being that the integral has most of its area near the stationary point t₀. The approximate value of the integral in the neighborhood of the stationary point is then obtained analytically by expanding $\phi(t)$ and f(t) in a Taylor series around t₀.