Talk:Method of steepest descent

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[edit] This page should split in two

I don't think "Method of Steepest Descent" and "Laplace's Method" should be the same page. While you could perhaps call Laplace's Method a special case, that is misleading. The Method of Steepest Descent is a technique to reduce a problem to a form where you can apply Laplace's Method. It would be much more valuable to have separate pages, so that there is some information on steepest descent and saddle-point methods (e.g. why deform a contour onto the steepest descent path, and why this has constant phase). Lavaka 19:41, 6 September 2006 (UTC)

[edit] fr link points to wrong article

The fr: link for this page points to the French article for 'Stationary Phase' -- not for 'Steepest Descent' -- I'm not sure how to fix this though as I don't know enough French to find the right article in French wikipedia. Help? Zero sharp 23:21, 26 November 2006 (UTC)

[edit] A couple of omissions

M is a large number

Large compared to what?

Let x0 be a global maximum of f(x), which, for simplicity, we will assume to be unique.

Ok, but the article should at least mention in passing how to deal with the case where it is not. --Starwed 09:54, 1 February 2007 (UTC)

Well, large number M here means that M\to\infty (that is made precise later in the article).
If the maximum is not unique, things can be complicated. If the maxima are separated from each other, say one here, and the other one more to the right, etc, one can apply this theorem to each individual maximum separately by splitting the interval of integration. If the maxima form an interval (say for a constant function) the theorem is not applicable. I'll reword the statement of the theorem a bit. Oleg Alexandrov (talk) 15:52, 1 February 2007 (UTC)
This is rather a belated reply, but it really is necessary to specify what large means in this context. For any given function, the method will be a useful approximation for some values of M, but break down as M gets smaller. --Starwed 03:36, 25 June 2007 (UTC)

A question involving perhaps divergent series, let be:

\int_{a}^{b} dx e^{Mf(x)}= g(M)(1+M^{-1}a1+a2M^{-2}+............)

Laplace in admitted in general that this series in 1/M would be divergent if M isn't big but could be summable (in Borel or other sense??)? --85.85.100.144 22:12, 16 February 2007 (UTC)