Stationary phase approximation

The stationary phase method is an approach for solving integrals analytically by evaluating the integrands in regions where they contribute the most. This method is specifically directed to evaluating oscillatory integrands, where the phase function of the integrand is multiplied by a relatively high value. In our case, this value corresponds to the frequency and thus our approximation is asymptotically exact as the frequency approaches $\infty$.

Integrals of the form

$$I(k)=\int_{-\infty}^{\infty} e^{ik\phi(t)}f(t)\; dt$$

are approximated asymptotically Zauderer (1989) when $k \rightarrow \infty$ by  

$$I(k) \approx e^{ik\phi (t_0)} f(t_0) e^{{\rm sign} (\phi''(... ...t[{{2\pi} \over {k \mid \phi''(t_0) \mid }} \right]^{1 \over 2}$$ (33)

where t₀ is the ``stationary point'' in which the derivative of the phase is zero. The approximation described in equation (33) assumes the second derivative is non-zero, which is the case here.

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