Comparisons of best approximations with Chebyshev expansions for functions with logarithmic endpoint singularities

Zhang, Xiaolong

doi:10.1007/s11075-023-01538-5

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2024

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“…Just a few of these approaches range from polynomial approximations [1,2] to solving integral equations [3] to rational approximations [4,5] to expansions in Jacobi or Chebyshev polynomials [6]. The latter are particularly prized by many workers [7][8][9].…”

Section: Introductionmentioning

confidence: 99%

The Fourier–Legendre Series of Bessel Functions of the First Kind and the Summed Series Involving 1F2 Hypergeometric Functions That Arise from Them

Straton

2024

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The Bessel function of the first kind JNkx is expanded in a Fourier–Legendre series, as is the modified Bessel function of the first kind INkx. The purpose of these expansions in Legendre polynomials was not an attempt to rival established numerical methods for calculating Bessel functions but to provide a form for JNkx useful for analytical work in the area of strong laser fields, where analytical integration over scattering angles is essential. Despite their primary purpose, one can easily truncate the series at 21 terms to provide 33-digit accuracy that matches the IEEE extended precision in some compilers. The analytical theme is furthered by showing that infinite series of like-powered contributors (involving 1F2 hypergeometric functions) extracted from the Fourier–Legendre series may be summed, having values that are inverse powers of the eight primes 1/2i3j5k7l11m13n17o19p multiplying powers of the coefficient k.

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Section: Introductionmentioning

confidence: 99%

The Fourier–Legendre Series of Bessel Functions of the First Kind and the Summed Series Involving 1F2 Hypergeometric Functions That Arise from Them

Straton

2024

Axioms

View full text Add to dashboard Cite

show abstract

Comparisons of best approximations with Chebyshev expansions for functions with logarithmic endpoint singularities

Cited by 1 publication

References 23 publications

The Fourier–Legendre Series of Bessel Functions of the First Kind and the Summed Series Involving 1F2 Hypergeometric Functions That Arise from Them

The Fourier–Legendre Series of Bessel Functions of the First Kind and the Summed Series Involving 1F2 Hypergeometric Functions That Arise from Them

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