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ON TAYLOR COEFFICIENTS OF ANALYTIC FUNCTION RELATED WITH EULER NUMBER

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dc.contributor.authorKostin, A. B.
dc.contributor.authorSherstyukov, V. B.
dc.contributor.authorКостин, Андрей Борисович
dc.date.accessioned2024-12-25T16:33:18Z
dc.date.available2024-12-25T16:33:18Z
dc.date.issued2022
dc.description.abstractWe consider a classical construction of second remarkable limit. We pose a question on asymptotically sharp description of the character of such approximation of the number e. In view of this we need the information on behavior of the coefficients in the power expansion for the function (Formula Presented) converging in the interval (Formula Presented). We obtain a recurrent rule regulating the forming of the mentioned coefficients. We show that the coefficients form a sign-alternating sequence of rational numbers (−1)nan, where n ∈ N ∪ {0} and a0 = 1, the absolute values of which strictly decay. On the base of the Faá di Bruno formula for the derivatives of a composed function we propose a combinatorial way of calculating the numbers an as n ∈ N. The original function f(x) is the restriction of the function f(z) on the real ray x andgt; −1 having the same Taylor coefficients and being analytic in the complex plane C with the cut along (−∞, −1]. By the methods of the complex analysis we obtain an integral representation for an for each value of the parameter n ∈ N. We prove that an → 1/e as n → ∞ and find the convergence rate of the difference an − 1/e to zero. We also discuss the issue on choosing the contour in the integral Cauchy formula for calculating the Taylor coefficients (−1)nan of the function f(z). We find the exact values of arising in calculations special improper integrals. The results of the made study allows us to give a series of general two-sided estimates for the deviation e−(1+x) 1/x consistent with the asymptotics s of f(x) as x → 0. We discuss the possibilities of applying the obtained statements © Kostin A.B., Sherstyukov V.B. 2022
dc.format.extentС. 70-85
dc.identifier.citationKostin, A. B. ON TAYLOR COEFFICIENTS OF ANALYTIC FUNCTION RELATED WITH EULER NUMBER / Kostin, A.B., Sherstyukov, V.B. // Ufa Mathematical Journal. - 2022. - 14. - № 3. - P. 70-85. - 10.54708/23040122_2022_14_3_70
dc.identifier.doi10.54708/23040122_2022_14_3_70
dc.identifier.urihttps://www.doi.org/10.54708/23040122_2022_14_3_70
dc.identifier.urihttps://www.scopus.com/record/display.uri?eid=2-s2.0-85137289617&origin=resultslist
dc.identifier.urihttp://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcAuth=Alerting&SrcApp=Alerting&DestApp=WOS_CPL&DestLinkType=FullRecord&UT=WOS:000858741400006
dc.identifier.urihttps://openrepository.mephi.ru/handle/123456789/28193
dc.relation.ispartofUfa Mathematical Journal
dc.titleON TAYLOR COEFFICIENTS OF ANALYTIC FUNCTION RELATED WITH EULER NUMBER
dc.typeArticle
dspace.entity.typePublication
oaire.citation.issue3
oaire.citation.volume14
relation.isAuthorOfPublication3070740c-c832-420c-a070-a880195523bc
relation.isAuthorOfPublication.latestForDiscovery3070740c-c832-420c-a070-a880195523bc
relation.isOrgUnitOfPublicationd19559ab-04cd-486a-ae8e-f40ccd36a1a6
relation.isOrgUnitOfPublication.latestForDiscoveryd19559ab-04cd-486a-ae8e-f40ccd36a1a6
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