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37-542: Markovian is an adjective that may describe: In probability theory and statistics, subjects named for Andrey Markov : A Markov chain or Markov process, a stochastic model describing a sequence of possible events The Markov property , the memoryless property of a stochastic process The Markovians, an extinct god-like species in Jack L. Chalker's Well World series of novels Markovian Parallax Denigrate , references

74-465: A i b i {\displaystyle \operatorname {K} _{i=n}^{\infty }{\tfrac {a_{i}}{b_{i}}}} part of the fraction by w n , instead of by 0, to compute the convergents. The convergents thus obtained are called modified convergents . We say that the continued fraction converges generally if there exists a sequence { w n ∗ } {\displaystyle \{w_{n}^{*}\}} such that

111-649: A i b i {\displaystyle f_{n}=\operatorname {K} _{i=1}^{n}{\tfrac {a_{i}}{b_{i}}}} are the convergents of the continued fraction, converges absolutely . The Śleszyński–Pringsheim theorem provides a sufficient condition for absolute convergence. Finally, a continued fraction of one or more complex variables is uniformly convergent in an open neighborhood Ω when its convergents converge uniformly on Ω ; that is, when for every ε > 0 there exists M such that for all n > M , for all z ∈ Ω {\displaystyle z\in \Omega } , It

148-432: A generalized continued fraction sets each nested fraction on the same line, indicating the nesting by dangling plus signs in the denominators: Sometimes the plus signs are typeset to vertically align with the denominators but not under the fraction bars: Pringsheim wrote a generalized continued fraction this way: Carl Friedrich Gauss evoked the more familiar infinite product Π when he devised this notation: Here

185-535: A mysterious series of Usenet messages See also [ edit ] Markov (surname) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Markovian . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Markovian&oldid=1091366664 " Category : Disambiguation pages Hidden categories: Short description

222-412: A similar procedure to choose another sequence { d i } to make each partial denominator a 1: where d 1 = ⁠ 1 / b 1 ⁠ and otherwise d n + 1 = ⁠ 1 / b n b n + 1 ⁠ . These two special cases of the equivalence transformation are enormously useful when the general convergence problem is analyzed. As mentioned in the introduction,

259-411: Is any infinite sequence of non-zero complex numbers we can prove, by induction, that where equality is understood as equivalence, which is to say that the successive convergents of the continued fraction on the left are exactly the same as the convergents of the fraction on the right. The equivalence transformation is perfectly general, but two particular cases deserve special mention. First, if none of

296-530: Is different from Wikidata All article disambiguation pages All disambiguation pages Andrey Markov Andrey Andreyevich Markov (14 June 1856 – 20 July 1922) was a Russian mathematician best known for his work on stochastic processes . A primary subject of his research later became known as the Markov chain . He was also a strong, close to master-level, chess player. Markov and his younger brother Vladimir Andreevich Markov (1871–1897) proved

333-588: Is now called Gauss's continued fractions . They can be used to express many elementary functions and some more advanced functions (such as the Bessel functions ), as continued fractions that are rapidly convergent almost everywhere in the complex plane. The long continued fraction expression displayed in the introduction is easy for an unfamiliar reader to interpret. However, it takes up a lot of space and can be difficult to typeset. So mathematicians have devised several alternative notations. One convenient way to express

370-407: Is sometimes necessary to separate a continued fraction into its even and odd parts. For example, if the continued fraction diverges by oscillation between two distinct limit points p and q , then the sequence { x 0 , x 2 , x 4 , ...} must converge to one of these, and { x 1 , x 3 , x 5 , ...} must converge to the other. In such a situation it may be convenient to express

407-444: Is still the basis of many modern proofs of convergence of continued fractions . In 1761, Johann Heinrich Lambert gave the first proof that π is irrational , by using the following continued fraction for tan x : Continued fractions can also be applied to problems in number theory , and are especially useful in the study of Diophantine equations . In the late eighteenth century Lagrange used continued fractions to construct

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444-408: Is zero, the infinite continued fraction is really just a finite continued fraction with n fractional terms, and therefore a rational function of a 1 to a n and b 0 to b n + 1 . Such an object is of little interest from the point of view adopted in mathematical analysis, so it is usually assumed that all a i ≠ 0 . There is no need to place this restriction on

481-448: The a i are zero, a sequence { c i } can be chosen to make each partial numerator a 1: where c 1 = ⁠ 1 / a 1 ⁠ , c 2 = ⁠ a 1 / a 2 ⁠ , c 3 = ⁠ a 2 / a 1 a 3 ⁠ , and in general c n + 1 = ⁠ 1 / a n + 1 c n ⁠ . Second, if none of the partial denominators b i are zero we can use

518-450: The partial denominators , and the leading term b 0 is called the integer part of the continued fraction. The successive convergents of the continued fraction are formed by applying the fundamental recurrence formulas : where A n is the numerator and B n is the denominator, called continuants , of the n th convergent. They are given by the three-term recurrence relation with initial values If

555-471: The Euclidean algorithm , a procedure for finding the greatest common divisor of two natural numbers m and n . That algorithm introduced the idea of dividing to extract a new remainder – and then dividing by the new remainder repeatedly. Nearly two thousand years passed before Bombelli (1579) devised a technique for approximating the roots of quadratic equations with continued fractions in

592-1133: The Markov brothers' inequality . His son, another Andrey Andreyevich Markov (1903–1979), was also a notable mathematician, making contributions to constructive mathematics and recursive function theory. Andrey Markov was born on 14 June 1856 in Russia. He attended the St. Petersburg Grammar School, where some teachers saw him as a rebellious student. In his academics he performed poorly in most subjects other than mathematics. Later in life he attended Saint Petersburg Imperial University (now Saint Petersburg State University ). Among his teachers were Yulian Sokhotski (differential calculus, higher algebra), Konstantin Posse (analytic geometry), Yegor Zolotarev (integral calculus), Pafnuty Chebyshev (number theory and probability theory), Aleksandr Korkin (ordinary and partial differential equations), Mikhail Okatov (mechanism theory), Osip Somov (mechanics), and Nikolai Budajev (descriptive and higher geometry). He completed his studies at

629-464: The " K " stands for Kettenbruch , the German word for "continued fraction". This is probably the most compact and convenient way to express continued fractions; however, it is not widely used by English typesetters. Here are some elementary results that are of fundamental importance in the further development of the analytic theory of continued fractions. If one of the partial numerators a n + 1

666-656: The candidate's examinations, and he remained at the university to prepare for a lecturer's position. In April 1880, Markov defended his master's thesis "On the Binary Square Forms with Positive Determinant", which was directed by Aleksandr Korkin and Yegor Zolotarev. Four years later in 1884, he defended his doctoral thesis titled "On Certain Applications of the Algebraic Continuous Fractions". His pedagogical work began after

703-395: The continued fraction converges if the sequence of convergents { x n } tends to a finite limit. This notion of convergence is very natural, but it is sometimes too restrictive. It is therefore useful to introduce the notion of general convergence of a continued fraction. Roughly speaking, this consists in replacing the K i = n ∞ ⁡

740-411: The continued fraction. See Chapter 2 of Lorentzen & Waadeland (1992) for a rigorous definition. There also exists a notion of absolute convergence for continued fractions, which is based on the notion of absolute convergence of a series: a continued fraction is said to be absolutely convergent when the series where f n = K i = 1 n ⁡

777-422: The defense of his master's thesis in autumn 1880. As a privatdozent he lectured on differential and integral calculus. Later he lectured alternately on "introduction to analysis", probability theory (succeeding Chebyshev, who had left the university in 1882) and the calculus of differences. From 1895 through 1905 he also lectured in differential calculus . One year after the defense of his doctoral thesis, Markov

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814-459: The general solution of Pell's equation , thus answering a question that had fascinated mathematicians for more than a thousand years. Lagrange's discovery implies that the canonical continued fraction expansion of the square root of every non-square integer is periodic and that, if the period is of length p > 1 , it contains a palindromic string of length p − 1 . In 1813 Gauss derived from complex-valued hypergeometric functions what

851-506: The governance". Markov was removed from further teaching duties at St. Petersburg University, and hence he decided to retire from the university. Markov was an atheist . In 1912, he responded to Leo Tolstoy 's excommunication from the Russian Orthodox Church by requesting his own excommunication. The Church complied with his request. In 1913, the council of St. Petersburg elected nine scientists honorary members of

888-402: The mid-sixteenth century. Now the pace of development quickened. Just 24 years later, in 1613, Pietro Cataldi introduced the first formal notation for the generalized continued fraction. Cataldi represented a continued fraction as with the dots indicating where the next fraction goes, and each & representing a modern plus sign. Late in the seventeenth century John Wallis introduced

925-467: The odd part x odd are given by and respectively. More precisely, if the successive convergents of the continued fraction x are { x 1 , x 2 , x 3 , ...} , then the successive convergents of x even as written above are { x 2 , x 4 , x 6 , ...} , and the successive convergents of x odd are { x 1 , x 3 , x 5 , ...} . If a 1 , a 2 ,... and b 1 , b 2 ,... are positive integers with

962-407: The original continued fraction as two different continued fractions, one of them converging to p , and the other converging to q . The formulas for the even and odd parts of a continued fraction can be written most compactly if the fraction has already been transformed so that all its partial denominators are unity. Specifically, if is a continued fraction, then the even part x even and

999-491: The partial denominators b i . When the n th convergent of a continued fraction is expressed as a simple fraction x n = ⁠ A n / B n ⁠ we can use the determinant formula to relate the numerators and denominators of successive convergents x n and x n − 1 to one another. The proof for this can be easily seen by induction . Base case Inductive step If { c i } = { c 1 , c 2 , c 3 , ...}

1036-406: The perspective of number theory, these are called generalized continued fraction. From the perspective of complex analysis or numerical analysis , however, they are just standard, and in the present article they will simply be called "continued fraction". A continued fraction is an expression of the form where the a n ( n > 0 ) are the partial numerators , the b n are

1073-449: The sequence of convergents { x n } approaches a limit , the continued fraction is convergent and has a definite value. If the sequence of convergents never approaches a limit, the continued fraction is divergent. It may diverge by oscillation (for example, the odd and even convergents may approach two different limits), or it may produce an infinite number of zero denominators B n . The story of continued fractions begins with

1110-413: The sequence of modified convergents converges for all { w n } {\displaystyle \{w_{n}\}} sufficiently distinct from { w n ∗ } {\displaystyle \{w_{n}^{*}\}} . The sequence { w n ∗ } {\displaystyle \{w_{n}^{*}\}} is then called an exceptional sequence for

1147-415: The standard unqualified use of the term continued fraction refers to the special case where all numerators are 1, and is treated in the article Simple continued fraction . The present article treats the case where numerators and denominators are sequences { a i } , { b i } {\displaystyle \{a_{i}\},\{b_{i}\}} of constants or functions. From

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1184-433: The term "continued fraction" into mathematical literature. New techniques for mathematical analysis ( Newton's and Leibniz's calculus ) had recently come onto the scene, and a generation of Wallis' contemporaries put the new phrase to use. In 1748 Euler published a theorem showing that a particular kind of continued fraction is equivalent to a certain very general infinite series . Euler's continued fraction formula

1221-530: The university and was later asked if he would like to stay and have a career as a mathematician. He later taught at high schools and continued his own mathematical studies. In this time he found a practical use for his mathematical skills. He figured out that he could use chains to model the alliteration of vowels and consonants in Russian literature. He also contributed to many other mathematical aspects in his time. He died at age 66 on 20 July 1922. In 1877, Markov

1258-545: The university. Markov was among them, but his election was not affirmed by the minister of education. The affirmation only occurred four years later, after the February Revolution in 1917. Markov then resumed his teaching activities and lectured on probability theory and the calculus of differences until his death in 1922. Continued Fractions Different fields of mathematics have different terminology and notation for continued fraction. In number theory

1295-480: Was appointed extraordinary professor (1886) and in the same year he was elected adjunct to the Academy of Sciences. In 1890, after the death of Viktor Bunyakovsky, Markov became an extraordinary member of the academy. His promotion to an ordinary professor of St. Petersburg University followed in the fall of 1894. In 1896, Markov was elected an ordinary member of the academy as the successor of Chebyshev . In 1905, he

1332-415: Was appointed merited professor and was granted the right to retire, which he did immediately. Until 1910, however, he continued to lecture in the calculus of differences. In connection with student riots in 1908, professors and lecturers of St. Petersburg University were ordered to monitor their students. Markov refused to accept this decree, and he wrote an explanation in which he declined to be an "agent of

1369-471: Was awarded a gold medal for his outstanding solution of the problem About Integration of Differential Equations by Continued Fractions with an Application to the Equation ( 1 + x 2 ) d y d x = n ( 1 + y 2 ) {\displaystyle (1+x^{2}){\frac {dy}{dx}}=n(1+y^{2})} . During the following year, he passed

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