Eksempler på brug af Approximants på Engelsk og deres oversættelser til Dansk
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Padé goes further,and arranges the approximants, expressed each in its lowest terms, into a table.
By 1908 Padé had written 41 papers,29 of which were on continued fractions and Padé approximants.
Ved 1908 Padé havde skrevet 41 papirer,hvoraf 29 var på fortsatte fraktioner og Padé approximants.The approximants which Padé introduced in this paper are now known as the Padé-Hermite approximants.
De approximants som Padé indført i dette papir er nu kendt som Padé-Hermite approximants.It would be fair to say that this work is the first systematic study of Padé approximants.
Det ville være rimeligt at sige, at dette arbejde er den første systematiske undersøgelse af Padé approximants.Padé's doctoral supervisor Hermite had used approximants and continued fractions in his work of 1873 on proving the transcendence of e.
Padé's ph.d. -vejleder Hermite havde benyttet approximants og fortsatte fraktioner i sit arbejde af 1873 om beviser de transcendens af E.Between these two contributions by Frobenius,Darboux had looked at Padé approximants of the exponential function.
Mellem disse to bidrag fra Frobenius,Darboux havde kigget på Padé approximants af den eksponentielle funktion.He continued to investigate approximants, and in 1894 he published a memoir in which he generalised the continued fraction algorithm which Hermite had studied in 1863 and again in 1893.
Han fortsatte med at undersøge approximants, og i 1894 han offentliggjort en memoir, hvor han generaliserede den fortsatte brøkdel algoritme, som Hermite havde undersøgt i 1863 og igen i 1893.He proved results on their general structure andalso clearly set out the connection between Padé approximants and continued fractions.
Han viste resultater på deres generelle struktur ogogså klart redegjort for forbindelsen mellem Padé approximants og fortsatte fraktioner.In 1899 Padé published another major work on Padé approximants which, as we noted above, looked in depth at approximants of the exponential function.
I 1899 Padé offentliggjort et andet stort arbejde på Padé approximants der, som vi har nævnt ovenfor, kigget grundigt på approximants af den eksponentielle funktion.The method continued to be used from time to time by various mathematicians,for example Kummer in 1837 used Padé approximants to sum series which only converged very slowly.
Den metode, fortsatte med at blive brugt fra tid til anden af forskellige matematikere,f. eks Kummer i 1837 brugt Padé approximants at summen serier som kun konvergerede meget langsomt.The existence of approximants was, of course, well-known before Padé, but no systematic examination of them had been made except by Frobenius, who determined the important relations which normally exist between them.
Eksistensen af approximants var naturligvis kendt før Padé, men ingen systematisk undersøgelse af dem havde fundet sted, undtagen ved Frobenius, der bestemmes den vigtige forbindelser, som normalt findes mellem dem.The first who seemed to realise the full significance of the method of Padé approximants was Lagrange in a paper of 1776 where he related them to continued fractions.
Det første der syntes at indse den fulde betydning af metoden for Padé approximants var Lagrange i et papir i 1776, hvor han relateret dem til fortsat fraktioner.Padé's doctoral supervisor Hermite had used approximants and continued fractions in his work of 1873 on proving the transcendence of e. How much of this earlier work was known to Padé is less obvious and he certainly seemed to be unaware of the contributions of Frobenius.
Padé's ph.d. -vejleder Hermite havde benyttet approximants og fortsatte fraktioner i sit arbejde af 1873 om beviser de transcendens af E. Hvor meget af dette tidligere arbejde var kendt Padé er mindre indlysende og han sikkert syntes at være uvidende af bidragene fra Frobenius.In his thesis Padé made the first systematic study of what we call today Padé approximants, which are rational approximations to functions given by their power series.
I sin afhandling Padé foretaget den første systematiske undersøgelse af, hvad vi kalder i dag Padé approximants, som er rationel tilnærmelser til funktioner givet ved deres magt serien.Although the theory of Padé approximants which he had developed in his thesis, and in many later papers, was not quick to be taken up by many other mathematicians, it did become well known after Borel presented Padé approximants in his 1901 book on divergent series.
Selv om teorien om Padé approximants som han havde udviklet i sin afhandling, og i mange nyere afhandlinger, var ikke hurtig til at blive taget op af mange andre matematikere, det var blevet kendt, efter BOREL præsenteret Padé approximants i hans 1901 bog om forskellige serier.Other contributions were made by Laguerre and Chebyshev.Padé's doctoral supervisor Hermite had used approximants and continued fractions in his work of 1873 on proving the transcendence of e.
Andre bidrag blev foretaget af Laguerre og Chebyshev.Padé's ph.d. -vejleder Hermite havde benyttet approximants og fortsatte fraktioner i sit arbejde af 1873 om beviser de transcendens af E.The existence of approximants was, of course, well-known before Padé, but no systematic examination of them had been made except by Frobenius, who determined the important relations which normally exist between them. Padé goes further, and arranges the approximants, expressed each in its lowest terms, into a table.
Eksistensen af approximants var naturligvis kendt før Padé, men ingen systematisk undersøgelse af dem havde fundet sted, undtagen ved Frobenius, der bestemmes den vigtige forbindelser, som normalt findes mellem dem. Padé går videre, og arrangerer approximants, udtrykt hver på sit laveste tal, i en tabel.Van Vleck, at a meeting of the American Mathematical Society in Boston in 1903 said(see for example):The existence of approximants was, of course, well-known before Padé, but no systematic examination of them had been made except by Frobenius, who determined the important relations which normally exist between them.
Van Vleck, på et møde i American Mathematical Society i Boston i 1903 sagde(se for eksempel):Eksistensen af approximants var naturligvis kendt før Padé, men ingen systematisk undersøgelse af dem havde fundet sted, undtagen ved Frobenius, der bestemmes den vigtige forbindelser, som normalt findes mellem dem.Padé approximants appear in Hankel's doctoral thesis Über eine besondere Classe der symmetrischen Determinanten, written in 1861, while in his thesis of 1870, supervised by Weierstrass, Frobenius discovered identies between the approximants which he developed more fully in a paper he published twenty years later.
Padé approximants vises i Hankel's ph.d. -afhandling Über eine BESONDERE Classe der symmetrischen Determinanten, skrevet i 1861, mens der i sin afhandling i 1870, overvåges af Weierstrass, Frobenius opdaget identies mellem approximants som han udviklede mere fuldt ud i et papir, han offentliggjorde tyve år senere.Although the theory of Padé approximants which he had developed in his thesis, and in many later papers, was not quick to be taken up by many other mathematicians, it did become well known after Borel presented Padé approximants in his 1901 book on divergent series. Padé had made other significant contributions, however, such as publishing an elementary algebra book and translating Klein 's Erlangen programme from German into French.
Selv om teorien om Padé approximants som han havde udviklet i sin afhandling, og i mange nyere afhandlinger, var ikke hurtig til at blive taget op af mange andre matematikere, det var blevet kendt, efter BOREL præsenteret Padé approximants i hans 1901 bog om forskellige serier. Padé havde gjort andre betydelige bidrag dog, såsom forlagsvirksomhed en elementær algebra bog og omsætte Klein's Erlangen programmet fra tysk til fransk.
