ABSTRACT
Since the Laplace approximation is available for the asymptotic approximation of integrals of real-valued functions, it is quite natural to seek a “Laplace type” approximation for integrals of complex-valued functions which arise through Fourier inversion. Integrals which arise in Fourier inversion formulas may be considered special cases of complex-valued integrals such as ∫ b
ψ(t) ξn(t) dt . (7.1)
We are interested in the asymptotic behaviour of such integrals as n → ∞. For real-valued integrals of the Laplace type, we discovered that largest asymptotic contribution to the value of such an integral comes locally from the point t0 where the integrand is maximised. However, we cannot speak directly about the maximum of a complex-valued function, and must seek an alternative. An early statement of this principle, due to Cauchy, Stokes, Riemann and Kelvin is the principle of stationary phase. Consider the asymptotic value of an integral of the form
β(t) ei nα(t) dt (7.2)
as n →∞. We shall suppose that α(t) and β(t) are real-valued functions of the real variable t. We can borrow terminology from physics and call α(t) the phase function. The principle of stationary phase states that the largest asymptotic contribution to the value of the integral in (7.2). as n → ∞ comes locally from the neighbourhood of the point t0 where the phase function α(t) is stationary-a point t0 where α′(t0) = 0. That this should be the case becomes intuitively clear when we consider the
real and imaginary parts of the integral in (7.2), namely ∫ b a
β(t) cos [nα(t)] dt , ∫ b a
β(t) sin [nα(t)] dt .
