History of asymptotic expansion of Laplace’s method between Laplace and Erdélyi

Question

In 1774 Laplace understood that $I≔∫\textrm{d}x \exp kf(x)$ for $k≫0$ can be estimated if one knows 2-jet of $f$ at its point of maximum (as $I₀ ≔ ∫\textrm{d}x \exp kf₀(x)$ with $f₀$ quadratic with the same 2-jet as $f$. — And $I₀$ is easy to calculate explicitly). In 1956 Erdélyi wrote (in his book) a formula improving Laplace’s $\lim_{k→∞} I/I₀ = 1$ to an asymptotic expansion in $1/k$ for $I/I₀$ (in terms of the higher jet of $f$ at its maximum). Unfortunately, I’m not able to find a lot of info on what has been happening during these 180 years in between!

The first such an asymptotic expansion I can find is an example done in the famous paper by Debye of 1909. There are two major sub-themes:

1. He covered one-step-more-advanced context (the steepest descent — when $f$ is allowed to be complex), but these “more advanced” parts were not new¹⁾.
2. He wrote down “the extra terms” of the asymptotic expansion in $1/k$.

¹⁾ He attributes these “more advanced” parts to Riemann’s posthumous publication of 1876 (of his notes of 1863) and Graf–Gubler’s book of 1898. (He did not know about Ph.D. thesis of Pavel Nekrasov of 1886.)

Well, the Debye’s paper is usually considered as the trail-blazing work in the method of steepest descent. From this I guess that it must be “the part (2)” which makes this paper the fundamental reference on this subject.

But his approach to (2) makes sense in the real case too! So below I ignore (1) completely: my questions are about what makes sense in purely-real cases too.

Question: was Debye indeed the first one to write these extra terms? 135 years between Laplace and Debye seems to be too long a gap — during these years a lot of mathematicians focused on approximate calculations…

Likewise: Have there been other significant attempts on this subject in 45 years between Debye and Erdélyi?²⁾

²⁾ For example: J.R. Airey³⁾ already in ’37 “jumped two steps ahead”: when tabulating special functions, he started to investigate “what is happening beyond superasymptotics”. (Essentially, these were traces of Dingle’s resurgence — what is now called hyperasymptotics…)

³⁾ … who is not G.B. Airy — although IIRC he worked with asymptotic expansions of Airy function as well!

Update: Split Debye’s results in two parts, and accentuate the interest to the “purely real case”.

Answer 1

The article The origin of the method of steepest descent covers the pre-Debye history, tracing the method back to a 1829 paper by Cauchy, "Mémoire sur divers points d’analyse". Contributions by Riemann (1863) and Nekrasov (1885) are also discussed.