Extended Hadamard expansions for the Airy functions

Abstract

A new series expansion for the the Airy function is presented here that stems from the method of steepest descents and can be related to the Hadamard expansions as presented in prevous works cited in the manuscript, and which is convergent for all values of the complex variable. Hadamard expansions were introduced as an extension of the method of steepest descents and are defined in terms of a large number of non-systematic integration path subdivisions. Unlike them, the expansions in the present work originate in the splitting of the steepest descent in a number of segments that is not only finite but very small, and which are defined on the basis of the location of the branch points. One of the segments reaches to infinity and this gives rise to the presence of upper incomplete Gamma functions. This is one of the most important differences with the Hadamard series as defined in the aforementioned references, where all the incomplete Gamma functions are of the lower type. The theoretical interest of the new series expansion is twofold. First of all, it shows how to convert an asymptotic series into a convergent one with a finite splitting of the steepest descent path. Secondly, the inverse of the phase function that is part of the Laplace-type equation is Taylor-expanded around branch points to produce Puiseux series when necessary. In addition to this, the proposed analysis shows again how the Stokes phenomenon for the Airy function is related to the transition of the steepest descent paths at $\arg z = \pm 2 π/3$ from one to two. In regard to its computational application, these series expansions require a relatively small number of terms for each of them to reach a very high precision.

Figures (9)

• Figure 1: Integration paths in \ref{['eq-ai2']} and \ref{['eq-ai3']}. The path $\mathcal{L}_{21}$ in equation \ref{['eq-bi1']} is the same as $\mathcal{L}_{12}$ but reversed.
• Figure 2: Sheets of the Riemann surface for the solutions in Remark \ref{['rem1']}. a) The three-sheeted Riemann surface has three finite branch points and the branch point at infinity. b) Detail of the branch point at $s=0$. The series expansion in Lemma \ref{['lema1']} runs from negative $s$ values on sheet 3, which corresponds to images in the branch given by $\alpha_{3}(s)$, to positive values on sheet 2, which has its image values in the branch given by $\alpha_{2}(s)$, as indicated by the dotted line. If the minus sign were selected in equation \ref{['eq-gn2']}, then the sheets would be traversed in the opposite direction (dashed line). The arrows indicate the fact that the branches are expanded around the central point $s=0$ and not the eventual direction of the integration through a path, which we will take from $s=-\infty$ to $s=\infty$.
• Figure 3: Sheets of the Riemann surface for the complex algebraic curve given by \ref{['eq-chvar2']} and $\arg z > 0$. a) This is the case for $w\neq e^{\, i 2\pi/3}$. The series in Lemma \ref{['lema1b']} are expansions around points $\alpha_{0}^{+}$ and $\alpha_{0}^{-}$. The radius of the convergence disks are given by the distance to the branch points at $t=-\frac{4}{3}$ and $t=-\frac{4}{3}(1+w^{3/2})$. b) For the case $w= e^{\, i 2\pi/3}$, $\alpha_{0}=\alpha_{0}^{+}$ is a branch point and we need a Puiseux series around it. The radius of convergence is given then by $\rho=\frac{4}{3}$. If we were to deal with the case of $\arg z > 0$, the labeling of the sheets would change to follow the corresponding branches, but not the topology.
• Figure 4: Integration segments. The regions of convergence for the series expansions of Lemmas \ref{['lema1']} to \ref{['lema1c']} are represented in the above part of the figure. The bottom part shows the selection of segments for the integration segments as they are defined to produce the addends in equation \ref{['eq-aiIexp']}.
• Figure 5: The convergence disks of the expansions around $t=0,-4/3,-4/3\,(1+w_{b}^{3/2})$ with $w_{b}=e^{\, i \varphi_{b}}$ and $\varphi_{b}=\frac{2}{3}\arctan(-7/8)$ are shown above. The expansion around $t=-\frac{4}{3} (s=0)$ corresponds to the expansion in Lemma \ref{['lema1']} and has a converge disk of radius $\rho=\frac{4}{3}$. The expansion around $t=0$ has a decreasing radius for its convergence disk as $\varphi \rightarrow \frac{2\pi}{3}$ due to the approach of the branch point at $t=-4/3\,(1+w^{3/2})$. When $\frac{2}{3}\arctan(-7/8)\le |\varphi| < \frac{2\pi}{3}$, the series expansion is done around the branch point instead of around $t=0$ as described in Lemma \ref{['lema1d']}.

Theorems & Definitions (16)

• Remark 1
• Lemma \oldthetheorem
• proof : Proof
• Lemma \oldthetheorem
• proof : Proof
• Lemma \oldthetheorem
• proof : Proof
• Lemma \oldthetheorem
• proof : Proof
• Remark 2