Core Concepts

This paper presents new, computationally efficient asymptotic expansions for associated Legendre and conical functions when the degree or order is large, which are uniformly valid for a wide range of arguments, including at singularities.

Abstract

Dunster, T. M. (2024). Simplified uniform asymptotic expansions for associated Legendre and conical functions. arXiv preprint arXiv:2410.03002.

This paper aims to derive simplified asymptotic expansions for associated Legendre and conical functions when either the degree (ν) or order (µ) is large. The goal is to provide expansions that are uniformly valid for a wide range of arguments, including at the singularity z=1, and have coefficients that are easy to compute.

The author utilizes Liouville-Green (LG) type expansions of exponential form to construct asymptotic expansions for the coefficient functions appearing in the solutions. This approach leads to simple expressions for the coefficients, involving recursively defined polynomials and the LG variable. The expansions are expressed in terms of modified Bessel functions and their analytic continuations.

- New asymptotic expansions for associated Legendre functions (P and Q) and their analytic continuations are derived for large ν with bounded µ, and for large µ with bounded ν.
- Expansions for conical functions, which are associated Legendre functions with complex degree ν=-1/2+iτ, are also presented for large τ.
- The expansions are uniformly valid for a wide range of complex arguments, including at the singularity z=1, and are expressed in terms of elementary functions and modified Bessel functions.
- The coefficients in the expansions are given by simple, explicit formulas involving recursively defined polynomials, making them computationally efficient.

The paper provides a new set of asymptotic expansions for associated Legendre and conical functions that are both accurate and computationally efficient. These expansions are uniformly valid for a wide range of arguments, overcoming limitations of previous approximations. The simplicity of the coefficient formulas makes these expansions particularly useful for numerical computations involving these functions.

This research contributes significantly to the field of special functions by providing improved asymptotic expansions for Legendre and conical functions. These functions have numerous applications in physics, engineering, and other scientific disciplines. The new expansions offer enhanced accuracy and computational efficiency, benefiting various applications involving these functions.

The paper focuses on the case where one parameter (ν or µ) is large while the other is bounded. Future research could explore expansions for cases where both parameters are large simultaneously, which presents a more complex scenario with coalescing turning points and poles. Additionally, extending the analysis to derive explicit error bounds for the new expansions would be beneficial.

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by T. M. Dunste... at **arxiv.org** 10-07-2024

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The simplified asymptotic expansions presented in the paper offer significant advantages in terms of both accuracy and efficiency compared to traditional methods for computing Legendre and conical functions, especially when dealing with very large parameter values. Here's a breakdown:
Accuracy:
Uniform Validity: Existing asymptotic expansions often suffer from limited domains of validity. They might break down near the singularity at z=1 or for specific ranges of the argument. The new expansions, however, are uniformly valid for a wider range of z, including the crucial singularity at z=1. This uniformity is a major advantage, ensuring accuracy even in regions where previous methods struggled.
Simple Pole Handling: The paper cleverly addresses the challenge posed by simple poles in the differential equation, which typically cause standard Liouville-Green (LG) expansions to fail. By employing modified Bessel functions and carefully constructed coefficient functions, the new expansions maintain accuracy even near these poles.
Explicit and Computable Coefficients: A key strength of the new expansions lies in their explicit and readily computable coefficients. Unlike previous methods that often involved complex nested integrations or cumbersome recursion relations, the coefficients here are expressed in terms of simple recursive polynomials and the LG variable. This makes them much easier to evaluate, especially for very large parameter values where computational cost becomes a concern.
Efficiency:
Fast Convergence: The asymptotic expansions exhibit rapid convergence as the parameter values increase. This means that fewer terms in the expansion are needed to achieve a desired level of accuracy, leading to faster computations.
Straightforward Implementation: The simplicity of the coefficients and the overall structure of the expansions make them relatively straightforward to implement numerically. This reduces the potential for coding errors and simplifies the computational process.
In summary: The simplified asymptotic expansions provide a more accurate and efficient way to compute Legendre and conical functions, particularly for large parameter values. Their uniform validity, ability to handle simple poles, and explicit, easily computable coefficients make them a powerful tool for applications where these functions are essential.

Yes, the techniques employed in the paper hold considerable promise for extension to other special functions exhibiting similar properties to Legendre and conical functions. The core principles that enable these simplified expansions are:
Liouville-Green (LG) Expansions of Exponential Form: The paper leverages LG expansions, but crucially expresses them in an exponential form. This allows for the coefficients to appear within the arguments of exponential, hyperbolic, or trigonometric functions, leading to simpler expressions for these coefficients.
Identification of Suitable Accompanying Functions: The choice of modified Bessel functions as accompanying functions is key. These functions naturally capture the behavior of the solutions near the simple poles of the differential equation, where standard LG expansions break down. For other special functions, identifying appropriate accompanying functions that accurately represent the solution behavior near singularities or turning points would be crucial.
Cauchy Integral Formula for Near-Singularity Evaluation: The paper utilizes the Cauchy integral formula to extend the domain of validity to regions very close to the singularity at z=1. This technique could be similarly applied to other special functions where direct evaluation of the expansions near singularities might be problematic.
Applicability to Other Functions:
Hypergeometric Functions: Hypergeometric functions, with their rich structure and connections to various differential equations, are prime candidates. The challenge would lie in finding suitable accompanying functions and adapting the coefficient derivation process to the specific forms of the hypergeometric differential equations.
Mathieu Functions: Mathieu functions, arising in problems with elliptical geometry, also exhibit singularities and parameter-dependent behavior. The techniques in the paper could potentially be adapted to derive simplified expansions for Mathieu functions, particularly for large parameter regimes.
Challenges and Considerations:
Complexity of Differential Equations: The success of this approach depends on the specific form and complexity of the differential equations governing the special functions. More intricate equations might require more sophisticated accompanying functions and coefficient derivations.
Choice of Accompanying Functions: Selecting appropriate accompanying functions that accurately capture the solution behavior near critical points is crucial. This choice might not always be obvious and could require careful analysis of the differential equation and its solutions.
Despite these challenges, the techniques presented in the paper provide a valuable framework and inspiration for deriving simplified asymptotic expansions for a broader class of special functions.

The improved asymptotic expansions for Legendre and conical functions have the potential to significantly impact various fields where these functions are fundamental, including:
Quantum Mechanics:
Atomic and Molecular Physics: Legendre polynomials are essential in describing the angular part of atomic orbitals and molecular wavefunctions. The new expansions could lead to more accurate and efficient calculations of electronic structures, energy levels, and transition probabilities, particularly for highly excited states or heavy atoms.
Scattering Theory: Conical functions arise in scattering problems involving spheroidal or conical boundaries. The improved expansions could enhance the accuracy and speed of simulations involving particle scattering off such geometries, relevant in areas like nuclear physics, condensed matter physics, and optics.
Electromagnetism:
Antenna Design and Analysis: Legendre and conical functions are crucial in modeling electromagnetic fields radiated by antennas with spherical or conical shapes. The new expansions could enable more precise and efficient antenna design optimization, particularly for high-frequency applications where large parameter values are common.
Electromagnetic Scattering: Similar to quantum scattering, these functions are essential in solving electromagnetic scattering problems involving objects with spherical or conical geometries. The improved expansions could lead to more accurate simulations of radar cross-sections, light scattering by particles, and other electromagnetic phenomena.
Geophysics:
Geopotential Modeling: Legendre polynomials are fundamental in representing the Earth's gravitational potential. The new expansions could contribute to more accurate geoid models, crucial for precise satellite positioning, navigation, and understanding the Earth's gravity field.
Seismic Wave Propagation: Conical functions are relevant in modeling seismic wave propagation in media with conical or spheroidal layers. The improved expansions could enhance the accuracy and efficiency of seismic data analysis and interpretation, potentially aiding in earthquake prediction and resource exploration.
Overall Impact:
Increased Accuracy: The improved expansions promise more accurate calculations and simulations in these fields, leading to a better understanding of physical phenomena and more reliable predictions.
Enhanced Computational Efficiency: The faster convergence and simpler coefficient evaluation offered by the new expansions translate to reduced computational costs, allowing researchers to tackle more complex problems or achieve higher precision in their simulations.
New Possibilities: The ability to accurately and efficiently compute these functions for very large parameter values opens up new avenues for exploration. It allows researchers to investigate regimes previously inaccessible due to computational limitations, potentially leading to new discoveries and insights.

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