Saddle point method: a rigorous proof?

Question

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..

Answer 1

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