Euler's 1760 paper on divergent series

Barbeau, Edward J.; Leah, P.J.

doi:10.1016/0315-0860(76)90030-6

Cited by 35 publications

(21 citation statements)

References 27 publications

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“…As for instance discussed in articles by Barbeau [6], Barbeau and Leah [7], Kozlov [97], and Varadarajan [112], already Euler had frequently used divergent series. Later, when the concept of convergence was better understood, Euler's admittedly somewhat informal treatment of divergent series was criticized, and a strong tendency emerged to ban divergent series completely from the realm of rigorous mathematics.…”

Section: A Divergent Seriesmentioning

confidence: 99%

On the analyticity of Laguerre series

Weniger¹

2008

J. Phys. A: Math. Theor.

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The transformation of a Laguerre seriesn (z) to a power series f (z) = ∑ ∞ n=0 γ n z n is discussed. Since many nonanalytic functions can be expanded in terms of generalized Laguerre polynomials, success is not guaranteed and such a transformation can easily lead to a mathematically meaningless expansion containing power series coefficients that are infinite in magnitude. Simple sufficient conditions based on the decay rates and sign patters of the Laguerre series coefficients λ (α) n as n → ∞ can be formulated which guarantee that the resulting power series represents an analytic function. The transformation produces a mathematically meaningful result if the coefficients λ (α) n either decay exponentially or factorially as n → ∞. The situation is much more complicatedbut also much more interesting -if the λ (α) n decay only algebraically as n → ∞. If the λ (α) n ultimately have the same sign, the series expansions for the power series coefficients diverge, and the corresponding function is not analytic at the origin. If the λ (α) n ultimately have strictly alternating signs, the series expansions for the power series coefficients still diverge, but are summable to something finite, and the resulting power series represents an analytic function. If algebraically decaying and ultimately alternating Laguerre series coefficients λ (α) n possess sufficiently simple explicit analytical expressions, the summation of the divergent series for the power series coefficients can often be accomplished with the help of analytic continuation formulas for hypergeometric series p+1 F p , but if the λ

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Section: A Divergent Seriesmentioning

confidence: 99%

On the analyticity of Laguerre series

Weniger¹

2008

J. Phys. A: Math. Theor.

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“…Here In any event, in spite of the inability of the brain to wrap itself around the notion of infinity, no one has ever doubted the veracity of Equation (2).…”

Section: The One Dimensional Infinite Seriesmentioning

confidence: 99%

“…Euler very cleverly states that "the same quantities that are less than zero can be considered to be greater than infinity" [2]. Modern mathematicians would define the infinity in question as negative infinity.…”

Section: J Wallis L Euler and Infinitymentioning

confidence: 99%

“…Since, therefore, we never reach the end of an infinite series, we never get besides to such a place where it is necessary to add that remainder; accordingly, this same remainder not only can be neglected, but also should be, because nowhere is a place for it found. [12] How misunderstood or disregarded? A footnote is added to L. Euler's Elements, probably by the French translator, and directly contradicts the first statement quoted above.…”

Section: Other Ways Of Representing Rational Numbers By Means Of One mentioning

confidence: 99%

“…He wrote about the matter in an article written in Latin published in 1760 in the acts of the Saint Petersburg academy when he was still at the Berlin academy [1]. To simplify the bibliographical trail, I take the liberty of simply quoting the English translation of the Latin original presented by E. J. Barbeau and P. J. Leah in a most helpful article [2].…”

Section: Excursus: J Wallis L Euler and The Number Circlementioning

confidence: 99%

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