Examples of using Continued fraction in English and their translations into Danish
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Stieltjes studied the continued fraction.
Stieltjes studeret den fortsatte fraktion.His work on continued fractions had already been awarded the Ormoy Prize of the Académie des Sciences in 1893.
Hans arbejde på fortsat fraktioner allerede havde fået tildelt Ormoy Prize af Académie des Sciences i 1893.Catalan published extensively on continued fractions and number theory.
In examining periodic solutions of differential equations Bendixson used methods based on continued fractions.
I behandlingen af periodiske løsninger af differentialligninger Bendixson anvendt metoder baseret på fortsat fraktioner.In 1655 he gave a continued fraction expansion of 4/π.
I 1655 gav han en fortsat brøkdel udvidelse af 4/ π.The thesis studied hyperelliptic andrelated integrals in continued fractions.
Afhandlingen undersøgt hyperelliptiske ogbeslægtede integrals i fortsat fraktioner.The convergents for the continued fraction of r all satisfy this.
Convergents for den fortsatte brøkdel af F alle opfylder dette.His work on continued fractions had already been awarded the Ormoy Prize of the Académie des Sciences in 1893. Recherches sur les fractions continues is described in as.
Hans arbejde på fortsat fraktioner allerede havde fået tildelt Ormoy Prize af Académie des Sciences i 1893. Recherches sur les Stieltjes worked on almost all branches of analysis, continued fractions and number theory.
Stieltjes arbejdet på næsten alle grene af analyse, fortsatte fraktioner og talteori.He also found a continued fraction expansion for the integral, the convergents of which involved the Laguerre polynomials.
Han har også fundet en brøkdel fortsatte ekspansion for integreret, convergents som indebar Laguerre polynomier.Fortsat hans matematiske arbejde Ramanujan undersøgt fortsatte fraktioner og divergerende serie i 1908.However Rawlins believes that a continued fraction method was used to calculate the value 11/83 while Fowler proposes that the anthyphairesis(or Euclidean algorithm) method was used see also.
Men Rawlins mener, at en fortsat brøkdel metode blev anvendt til at beregne værdien 11/ 83, mens Fowler foreslår, at anthyphairesis(eller euklidisk algoritme) metode blev anvendt se også.Minding also worked on differential equations,algebraic functions, continued fractions and analytic mechanics.
Minding også arbejdet med differentialligninger,algebraiske funktioner, fortsatte fraktioner og analytisk mekanik.It deals with the development into a continued fraction of the generating function of a sequence satisfying a difference equation.
Den beskæftiger sig med udvikling til en fortsat brøkdel af de genererer funktion af en sekvens opfylder en forskel ligning.However after Brouncker correctly computed the first 10 places in the decimal expansion of π using his continued fraction expansion, Huygens accepted the result.
Men efter Brouncker korrekt beregnet de første 10 steder i decimal udvidelse af π ved hjælp af hans fortsatte brøkdel ekspansion, Huygens accepteret resultatet.Khinchin first published the book Continued Fractions in 1936 with a second edition being published in 1949.
Khinchin første gang offentliggjort bogen Fortsat Brøker i 1936 med en anden udgave blev offentliggjort i 1949.He published A method of finding the sum of several powers of the sines in 1793 and in a paper of 12 August 1795 he gave an expansion of log10 n, where n a positive number,as a simple continued fraction and then computed log10 99 to nine decimal places.
Han offentliggjorde en metode til at finde summen af flere beføjelser til Sines i 1793 og i et papir af 12 august 1795 gav han en udvidelse af log 10 n,hvor n et positivt tal, som en simpel fortsatte fraktion og derefter beregnet log 10 99 til ni decimaler.Other topics which Heawood wrote on were continued fractions, approximation theory, and quadratic residues.
Andre emner som Heawood skrev den blev videreført fraktioner, tilnærmelse teori, og kvadratisk restprodukter.In 1655 he gave a continued fraction expansion of 4/π This result, written up in around ten pages, was added by Wallis to his treatise Arithmetica Infinitorum and probably first discovered by Brouncker in 1654.
I 1655 gav han en fortsat brøkdel udvidelse af 4/ π Dette resultat, der er skrevet op i omkring ti sider, blev tilføjet ved Wallis til hans afhandling Arithmetica Infinitorum og sandsynligvis første opdaget af Brouncker i 1654.The work represents the first general treatment of continued fractions as part of complex analytic function theory;
Det arbejde udgør den første generelle behandling af fortsatte fraktioner som en del af komplekse analytiske funktion teori;In fact it was Pringsheim in 1898 who first noted that Lambert's proof was absolutely correct and exceptional for its time, since the expansion of the tangent function was not only written down formally, butalso proved to be a convergent continued fraction.
I virkeligheden var det Pringsheim i 1898 der første bemærkes, at Lambert's bevis var helt korrekt og usædvanlige for sin tid, da udvidelsen af den tangent funktion ikke kun var skrevet ned formelt, menogså vist sig at være en konvergerende fortsatte fraktion.If we consider the first n terms of this continued fraction then we obtain the rational function P n(z)/Q n z.
Hvis vi ser på de første n vilkårene i denne fortsatte brøkdel så vi opnår en rationel funktion P n(z)/ Q n z.Among a long list of other results we mention just a very few such as his generalisation of Wolstenholme 's theorem; his work on classes of quintics not soluble by radicals; his closed form for the Bernoulli numbers; and his work on the length of the period of the continued fraction expansion of√N.
Blandt en lang liste af andre resultater, vi blot nævne nogle få såsom hans generalisering af Wolstenholme's sætning; hans arbejde med klasser af quintics ikke opløseligt ved radikaler, hans lukkede form for Bernoulli numre, og hans arbejde med længden af He wrote his doctoral thesis on algorithms for continued fractions which he submitted to the University of St Petersburg.
Han skrev sin ph.d. -afhandling om algoritmer for fortsat fraktioner som han sendte til universitetet i St. Petersborg.He was a pioneer in the application of mechanical methods, including digital computers, to the solution of problems in number theory and he talked about some of the methods used to factorise numbers including: factor tables, trial division, Legendre 's method,factor stencils, the continued fraction method, Fermat 's method, methods based on quadratic forms, and Shanks' method.
Han var en pioner i anvendelsen af mekaniske metoder, herunder digitale computere, til løsningen af problemer i talteori og han talte om nogle af After 1900 Markov applied the method of continued fractions, pioneered by his teacher Pafnuty Chebyshev, to probability theory.
Efter 1900 Markov anvendt metoden for fortsat fraktioner, banebrydende som hans lærer Pafnuty Chebyshev, at sandsynligheden teori.This result, written up in around ten pages, was added by Wallis to his treatise Arithmetica Infinitorum and probably first discovered by Brouncker in 1654. Wallis told Huygens of this result and Huygens expressed strong doubts that it was true.However after Brouncker correctly computed the first 10 places in the decimal expansion of π using his continued fraction expansion, Huygens accepted the result.
Dette resultat, der er skrevet op i omkring ti sider, blev tilføjet ved Wallis til hans afhandling Arithmetica Infinitorum og sandsynligvis første opdaget af Brouncker i 1654. Wallis fortalte Huygens af dette resultat og Huygens udtrykt stærk tvivl om, atdet var sandt. Men efter Brouncker korrekt beregnet de første 10 steder i decimal udvidelse af π ved hjælp af hans fortsatte brøkdel ekspansion, Huygens accepteret resultatet.His thesis was on Stieltjes continued fraction expansions, a topic which he continued to develop in his later work.
Hans tese var på Stieltjes fortsatte brøkdel udvidelser, et emne, som han He continued to investigate approximants, andin 1894 he published a memoir in which he generalised the continued fraction algorithm which Hermite had studied in 1863 and again in 1893.
Han In 1733 de Lagny examined the continued fraction expansion of the quotient of two integers and, as an example, considered adjacent Fibonacci numbers as the worst case expansion for the Euclidean algorithm.
I 1733 de Lagny undersøgt den fortsatte brøkdel udvidelse af kvotienten af to heltal og, som et eksempel, betragtes tilstødende Fibonacci tal som i værste fald ekspansion for euklidisk algoritme.
