# Saddle point method: a rigorous proof?

I am trying to prove in a fully rigorous way the Saddle Point method for holomorphic functions of 1 complex variable.

In books I find only complicated general statements or non-rigorous proofs. Hence I am following the proof in http://en.wikipedia.org/wiki/Method_of_steepest_descent , with the simplification that $S,f:\mathbb{C}\to\mathbb{C}$ (i.e. n=1).

I don't understand the step (11), inside the proof of equation (8). What is the rigorous argument for truncating the Taylor expansion at the term of order zero?

Alternatively can you give me any reference for a simple and rigorous proof? The reference cited in Wikipedia is Fedoryuk, but his book is in Russian and unfortunately I can't read Russian..

• I almost want to say "simple or rigorous: choose one." I can recommend some books containing rigorous proofs of the method but I don't know if they'd necessarily meet the "simple" criteria. Apr 18 '14 at 15:13
• I am looking for a rigorous proof. But strong hypothesis and weak results are ok (for example the statement here en.wikipedia.org/wiki/Method_of_steepest_descent only for the case n=1) Apr 18 '14 at 15:27
• Try de Bruijn's Asymptotic Methods in Analysis and Miller's Applied Asymptotic Analysis. Apr 18 '14 at 15:31
• Thank you for the references. I found the second book, but I don't see a stated and proved theorem. I would like to avoid all the things about the path of steepest descent and just use the following simple hypothesis: the path passes trough a non-degenerate saddle point $z_0$ for $S$ and the real part of $S$ resctricted to the path has its only global maximum point in $z_0$. Apr 18 '14 at 16:41
• I left my copy of the book in my office so unfortunately I can't give you a specific page number, but section 4.4 deals with saddle points. Ultimately the whole of chapter 4 describes the saddle point method. Apr 18 '14 at 16:44

## 1 Answer

Less known but wonderful book:

J. Dieudonné, Calcul infinitésimal, Hermann, 1970;

or, more recent and in English, the introduction of:

"The saddle-point method in C^N and the generalized Airy functions", DOI: 10.24033/bsmf2780