Identities in the spirit of Euler

Sofo, Anthony

doi:10.17398/2605-5686.35.1.21

Cited by 1 publication

(3 citation statements)

References 17 publications

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“…Equation ( 15) is a generalization of Equation (8). Consistency between these two equations can be observed by taking M = 1.…”

Section: Derivationmentioning

confidence: 99%

“…In particular, the substitution of M = 1 into series expansion (15) of the arctangent function implies that t = γ 1,1 = 1/2. Consequently, from Equation ( 15) we obtain Equation (8), where the expansion coefficients are…”

Section: Derivationmentioning

confidence: 99%

“…Many new identities and series expansions related to the arctangent function have been reported [7][8][9][10][11][12][13][14][15]. This shows that the discovery of new equations related to the arctangent function as well as their applications remains a very interesting topic.…”

Section: Introductionmentioning

confidence: 97%

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A Generalized Series Expansion of the Arctangent Function Based on the Enhanced Midpoint Integration

et al. 2023

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In this work, we derive a generalized series expansion of the acrtangent function by using the enhanced midpoint integration (EMI). Algorithmic implementation of the generalized series expansion utilizes a two-step iteration without surd or complex numbers. The computational test we performed reveals that such a generalization improves the accuracy in computation of the arctangent function by many orders of magnitude with increasing integer M, associated with subintervals in the EMI formula. The generalized series expansion may be promising for practical applications. It may be particularly useful in practical tasks, where extensive computations with arbitrary precision floating points are needed. The algorithmic implementation of the generalized series expansion of the arctangent function shows a rapid convergence rate in the computation of digits of π in the Machin-like formulas.

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“…Equation ( 15) is a generalization of Equation (8). Consistency between these two equations can be observed by taking M = 1.…”

Section: Derivationmentioning

confidence: 99%

Section: Derivationmentioning

confidence: 99%