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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Generalized continued fraction ： ウィキペディア英語版 Generalized continued fraction In complex analysis, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denominators can assume arbitrary real or complex values. A generalized continued fraction is an expression of the form :$x\; =\; b\_0\; +\; \backslash cfrac\}\}\}$ where the ''a''''n'' (''n'' > 0) are the partial numerators, the ''b''''n'' are 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: :$$ x_0 = \frac = b_0, \qquad x_1 = \frac = \frac,\qquad x_2 = \frac = \frac,\qquad\cdots\, and in general〔Jones & Thron (1980) p.20〕 :$$ A_n = b_n A_ + a_n A_, \qquad B_n = b_n B_ + a_n B_, \, where ''A''''n'' is the ''numerator'' and ''B''''n'' is the ''denominator'', called continuants, of the ''n''th convergent. If the sequence of convergents 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''. == History == The story of continued fractions begins with the Euclidean algorithm,〔300 BC Euclid, ''Elements'' - The Euclidean algorithm generates a continued fraction as a by-product.〕 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 Rafael Bombelli〔1579 Rafael Bombelli, ''L'Algebra Opera''〕 devised a technique for approximating the roots of quadratic equations with continued fractions. Now the pace of development quickened. Just 24 years later Pietro Cataldi introduced the first formal notation〔1613 Pietro Cataldi, ''Trattato del modo brevissimo di trovar la radice quadra delli numeri''; roughly translated, ''A treatise on a quick way to find square roots of numbers''.〕 for the generalized continued fraction. Cataldi represented a continued fraction as :$a\_0.\backslash ,$ &$n\_1\; \backslash over\; d\_1.$ &$n\_2\; \backslash over\; d\_2.$ &$,$ with the dots indicating where the next fraction goes, and each & representing a modern plus sign. Late in the seventeenth century John Wallis〔1695 John Wallis, ''Opera Mathematica'', Latin for ''Mathematical Works''.〕 introduced 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.〔1748 Leonhard Euler, ''Introductio in analysin infinitorum'', Vol. I, Chapter 18.〕 Euler's continued fraction formula is still the basis of many modern proofs of convergence of continued fractions. In 1761, Johann Heinrich Lambert gave the first proof that is irrational, by using the following continued fraction for :〔''The Irrationals: A Story of the Numbers You Can't Count On'', Julian Havil, Princeton University Press, 2012, pp.104-105〕 :$\backslash tan(x)\; =\; \backslash cfrac\backslash ddots\}\}\}\}$ 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 the general solution of Pell's equation, thus answering a question that had fascinated mathematicians for more than a thousand years.〔Brahmagupta (598 - 670) was the first mathematician to make a systematic study of Pell's equation.〕 Amazingly, 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 is now called Gauss's continued fractions.〔1813 Carl Friedrich Gauss, ''Werke'', Vol. 3, pp. 134-138.〕 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Generalized continued fraction」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.