Talk:Method of steepest descent

Article history

This article started with content mostly focused on Laplace's method that was moved on April 25, 2010 to the article Laplace's method (see discussion on talk:Laplace's method for details). Laplace's method article still retains some text and formulas that are specific to the Method of steepest descent, that, i believe, would be better placed within this article (with links from there to here). Indeed, there is a lot of room for improvements in both articles. Have fun. --Bluemaster (talk) 19:52, 26 April 2010 (UTC)

Examples

It would be great to have some examples of how the method of steepest descent can be used to evaluate integrals. (talk) 3:02 Aug 17, 2010 (UTC)

Consistency check and proof reading are wanted

I have substantially enlarged this article by adding sections 1-4, which are largely based on my personal notes. Since I am not a mathematician (I am a physicist), I think proof reading is much needed by more experienced people. Denysbondar (talk) 02:29, 3 July 2012 (UTC)

Wild West?

"posses a degenerate saddle point," doesn't look quite right to me. But I'm afraid I don't understand the subject well enough to know whether "poses a degenerate saddle point," or "possesses a degenerate saddle point," would be more appropriate. ϢereSpielChequers 11:53, 19 August 2012 (UTC)

Arrow diagrams misleading

The diagram "An illustration of complex Morse lemma" could be improved by changing the arrow to point from $w$ to $\phi(w)$. — Preceding unsigned comment added by Patrick12345678 (talk • contribs) 23:43, 21 January 2013 (UTC)

A good suggestion! I have altered the figure. Denysbondar (talk) 17:19, 27 January 2013 (UTC)

The asymptotic expansion in the case of a single non-degenerate saddle point

Concerning the section about the case of a single non-degenerate saddle point, I'm surprised by the second condition, which requires that the real part of $S$ (a holomorphic function) has a single maximum. As far as I know, the image of an open set under a non-constant holmorphic function is an open set, and the projection onto the real line is still going to be open, hence no maximum. What gives? I suspect it could be that there is a unique maximum in $I_{x}$, but then we are not talking about a saddle-point approximation, simply Laplace's method for real integrals, no?107.6.61.163 (talk) 14:45, 5 July 2013 (UTC)

In this section, why do we ask for x_0 to live in I_x? In the figure, x_0 is apparently live in \Omega_x, that is why we need to deform the countour, right?

This is REALLY REALLY complicatedly explained

I learned MSD in my math class last month. I can't make heads or tails of this article, and I think I understand this concept pretty well, at least as the text Advanced Mathematical Methods for Physicists explains it. Any tips on rewriting the intro/first example for clarity and less confusing technicality? I'm pretty new to Wikipedia, so any pointers on style would be welcome.

Laudiacay (talk) 17:44, 11 April 2016 (UTC)

I agree with you. I just finished reading the explanation of this method in the second chapter of Wong's "Asymptotic Approximation of Integrals". The lucidity of the exposition in that book is hard to find in this article. Interestingly, Wong himself remarks that the available description of this method is the most sketchiest and as a result "is probably also the most difficult to understand among the classical methods." The second chapter does not mention the Morse lemma at all, until much later in chapter nine for multi-dimensional problems. The article could have been written from the perspective of one dimensional functions first, and then extended for higher dimensions. Also an archetypal example of the Airy integral could have been provided. As it stands, the article fails to be useful to a casual curious readers with moderate background in math. Manoguru (talk) 23:50, 24 July 2017 (UTC)