Up to this point I have got a grasp of some basics of "steepest descent method" to evaluate the integral of a complex exponential function ##f(z) = \exp(A(x,y))\exp(iB(x,y))##. Using this method the original integration path is modified in such a way that it passes through its saddle points, assuming this function is analytic everywhere. The modified path coincides with the line on which the magnitude of ## f(z) ##, namely ##\exp(A(x,y))##, is changing most rapidly.(adsbygoogle = window.adsbygoogle || []).push({});

But then I also saw there is another method to calculate the integral of the same function, "stationary phase method". I haven't gone through the related literature though, but judging from the name I suspect both methods are identical. This is because in steepest descent method one follows the line of the most rapid change of ##A(x,y)##, which is also the line of constant phase (constant ##B(x,y)##), for them being analytic. Now my intuition says that stationary phase means constant phase, so does this mean that in "stationary phase method" one also follows the same path as that in "steepest descent method"? If so, what is the difference between both methods? And in which cases one method is better than the other?

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# Steepest descent vs. stationary phase method

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