Convergence Acceleration of Infinite Series Involving the Product of Riccati–Bessel Function and Its Application for the Electromagnetic Field: Using the Continued Fraction Expansion Method

A summation technique has been developed based on the continuous fractional expansion to accelerate the convergence of infinite series involving the product of Riccati–Bessel functions, which are common to electromagnetic applications. The series is transformed into a new and faster convergent seque...

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Bibliographic Details
Published in:Applied Computational Electromagnetics Society journal 2022-03, Vol.36 (12), p.1518
Main Authors: Fanghua, Zheng, Qingyun, Di, Zhe, Yun, Ya, Gao
Format: Article
Language:English
Subjects:
Acceleration
Addition theorem
Algorithms
Bessel functions
Convergence
Electric dipoles
Electromagnetic fields
Electromagnetism
Infinite series
Mathematical analysis
Mie scattering
New technology
Spherical harmonics
Spherical waves
Wave functions
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Summary:A summation technique has been developed based on the continuous fractional expansion to accelerate the convergence of infinite series involving the product of Riccati–Bessel functions, which are common to electromagnetic applications. The series is transformed into a new and faster convergent sequence with a continued fraction form, and then the continued fraction approximation is used to accelerate the calculation. The well-known addition theorem formula for spherical wave function is used to verify the correctness of the algorithm. Then, some fundamental aspects of the practical application of continuous fractional expansion for Mie scattering theory and electromagnetic exploration are considered. The results of different models show that this new technique can be applied reliably, especially in the electromagnetic field excited by the vertical electric dipole (VED) source in the “earth-ionospheric” cavity. The comparison among the new technology, the Watson-transform, and the spherical harmonic series summation algorithm shows that this new technology only needs less than 120 series items which is already enough to obtain a small relative error, which greatly improves the convergence speed, and provides a new way to solve the problem.
ISSN:1054-4887
1943-5711
DOI:10.13052/2021.ACES.J.361202