Continued Fractions Vol 1: Convergence Theory by Lisa Lorentzen

By Lisa Lorentzen

Persisted Fractions contains volumes -- quantity 1: Convergence idea; and quantity 2: illustration of features (tentative title), that is anticipated in 2011. quantity 1 is devoted to the convergence and computation of endured fractions, whereas quantity 2 will deal with representations of meromorphic capabilities by way of persevered fractions. Taken jointly, the 2 volumes will current the fundamental endured fractions conception with out requiring an excessive amount of prior wisdom; a few simple wisdom of advanced capabilities will suffice. either new and complicated graduate scholars of persevered fractions shall get a finished realizing of ways those countless buildings paintings in a few functions, and why they paintings so good. A various buffet of attainable purposes to whet the urge for food is gifted first, earlier than the extra easy yet modernized idea is given.

This re-creation is the results of an expanding curiosity in computing unique services via persisted fractions. The equipment defined intimately are, in lots of situations, extremely simple, but trustworthy and effective.

Contents:

  • Introductory Examples;
  • Basics;
  • Convergence standards;
  • Periodic and restrict Periodic persevered Fractions;
  • Numerical Computation of persisted Fractions, a few endured Fraction Expansions.

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Extra info for Continued Fractions Vol 1: Convergence Theory

Example text

Use the procedure of Example 13 on page 30 to find the first terms a1 z a2 z a3 z a4 z 1 + 1 + 1 + 1 +··· of a continued fraction expansion of f (z) := ez − 1. Problems 51 22. From continued fraction to power series. Use the procedure of Example 14 on page 33 to find the first terms of the power series expansion at 0 corresponding to the continued fraction z −z/2 z/6 −z/6 1 + 1 + 1 + 1 +· · · 23. ♠ From approximants to continued fraction. Let {fn }∞ n=0 with f0 := 0 and fn = fn−1 for all n be a given sequence of complex numbers.

A more primitive way is the method of successive substitutions ([Lamb61]) due to Lambert (1728 1777), a colleague of Euler and Lagrange in Berlin. We illustrate this method by an example: Example 13. We shall compute the circumference L of the ellipse y2 x2 + = 1, a2 b2 a ≥ b ≥ 0, a > 0. 2) The well known arc length formula leads to the elliptic integral π/2 L = 4a 0 where ε := By setting 1 − ε2 sin2 θ dθ √ a2 − b2 /a is the eccentricity of the ellipse, and thus b2 = a2 (1 − ε2 ). 4) 2 2 2 2 2 ([H¨ utte55], [LoWa85]).

Khovanskii’s book ([Khov63]) contains some interesting applications. The history of continued fractions up to 1939 is described in Brezinski’s book ([Brez91]). Also the texts mentioned above, in particular the book by Jones and Thron, contain interesting comments on the historic development of concepts, methods and applications. For the computational aspects of continued fractions we recommend the handbook ([CJPVW7]). To most people (meaning mathematicians) continued fractions are the regular continued fractions in number theory.

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