Article In: orcid
On Asymptotic Series for Generalized Airy, Circular, and Hyperbolic Functions
Mathematics
— 2025 — MDPI
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Authors:
Published in
December 23, 2025
Abstract
The paper concerns the solution of the ordinary differential equation 𝑦″ ±𝑥𝑚𝑦 =0, which may be designated the generalized Airy equation, since the original Airy equation corresponds to the particular case 𝑚 =1 with the + sign. The solutions may be designated generalized circular (hyperbolic) sines and cosines for the + (−) sign, since the particular case 𝑚 =0 corresponds to the elementary circular (hyperbolic) sines and cosines. There are 3 cases of solution of the generalized Airy equation, depending on the parameter m: (I) for m a non-negative integer, the coefficient 𝑥𝑚 is an analytic function, and the solutions are also analytic series; (II) for m complex other than an integer, the coefficient 𝑥𝑚 has a branch point at the origin, and the solutions also have a branch point multiplied by an analytic series; (III) for m a negative integer, the coefficient 𝑥𝑚 has a pole of order m, and the generalized Airy equation is singular. Case III has four subcases: (III-A) for 𝑚 =−1, the coefficient 𝑥−1 is a simple pole, and the solutions are Frobenius–Fuchs series of two kinds; (III-B) for 𝑚 =−2, the coefficient is a double pole, and the solutions are a combination of elementary functions, namely exponential, logarithmic, and circular (hyperbolic) sine and cosine for the + (−) sign; (III-C,D) for 𝑚 =−3,−4,…, the coefficient is a pole of multiplicity |𝑚|, and the generalized Airy differential equation has an irregular singularity of degree |𝑚| −2 at the origin. In the sub-cases (III-C,D), the solutions can be obtained by inversion as asymptotic series of descending powers specified by (III-C) Frobenius–Fuchs series of two kinds for a triple pole 𝑚 =−3; (III-D) for higher-order poles 𝑚 =−4,−5,… by generalized circular (hyperbolic) sines and cosines of 1/𝑥. It is shown that in all cases the ascending and descending series are absolutely and uniformly convergent with the n-th term decaying like 𝑂(𝑛2). This enables the use of a few terms of the series to obtain tables and plot graphs of the solutions of the generalized Airy differential equation as generalized circular and hyperbolic sines and cosines for several values of the parameter m. As a physical application, it is shown that the generalized circular (hyperbolic) cosines and sines specify the motion of a linear oscillator with natural frequency a power of time in the oscillatory (monotonic) case when the origin is an attractor (repeller).
Publication details
Authors in the community:
Manuel José dos Santos Silva
ist178213
Publication version
CVoR - Corrected Version of Record
Publisher
MDPI
Link to the publisher's version
https://www.mdpi.com/
Title of the publication container
Mathematics
First page or article number
52
Volume
14
Issue
1
ISSN
2227-7390
Fields of Science and Technology (FOS)
mathematics - Mathematics
Keywords
- generalized circular and hyperbolic functions
- Airy functions
- Frobenius–Fuchs series
- singularities
- asymptotic series
Publication language (ISO code)
eng - English
Alternative identifier (URI)
https://www.mdpi.com/2227-7390/14/1/52
Rights type:
Only metadata available
Creative Commons license
CC-BY - Attribution
Financing entity
Fundação para a Ciência e a Tecnologia
Identifier for the funding entity: https://doi.org/10.13039/501100001871
Type of identifier of the funding entity: Crossref Funder
Number for the project, award or grant: UID/50022/2025