The Method of Stationary Phase
Consider evaluating the following integral in the limit X ^ ro.
1,X) = и, х)е"»<Ъ.
It was pointed out by Lord Kelvin (1887) that for large X the function еіЛФ(х) will oscillate wildly with almost complete cancellation. Significant contribution to the integral comes from small intervals of х where Ф(х) varies slowly. This happens when Ф'(х) = 0 or at the stationary points of Ф(х).
Suppose Ф(х) has only one stationary point in (a, b), say at х = с. That is
Ф, х) = Ф, с) +—— 2C, х – c2 ) + •••• (B2)
Since the major contribution to (B1) comes around х = с, asymptotically the integral is given by
И, с)е1ХФ, с) еФ" ,c),x-c)2Dx.
It is possible to change the limits of integration to – ro to ro as the error of extending 5 to ro is asymptotically small (nearly complete cancellation). On accounting for that Ф"(с) may be positive or negative, the integral of (B3) may be evaluated to give
lim И, х)еаФ, х)ёх – — 2 И, cyH(c)+sgn^"(c))f], (B4)
x->ro w X Ф, c)| w ()
where sgn( ) is the sign of ( ).