Continued Fractions: A Modern and Classical Journey Into the World of Siegel's Continued Fractions
暫譯: 連分數：西格爾連分數的現代與古典之旅

Elsner, Carsten, Havens, Christopher Robin

Continued Fractions: A Modern and Classical Journey Into the World of Siegel's Continued Fractions
  • 出版商: Springer
  • 出版日期: 2025-10-04
  • 售價: $6,220
  • 貴賓價: 9.5 折 $5,909
  • 語言: 英文
  • 頁數: 366
  • 裝訂: Hardcover - also called cloth, retail trade, or trade
  • ISBN: 3031995457
  • ISBN-13: 9783031995453
  • 相關分類: 離散數學 Discrete-mathematics

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商品描述

This monograph originates from a study of the continued fraction [1, 2, 3, ...], which we call the Zopf number. Its origins date back to 1929 when Siegel introduced it as a ratio of Bessel functions. Continued fractions is most often styled classically, and much of the content is formulated through Diophantine analysis. However, in this book aspects of the theory of computation can be used interchangeably through matrices and transducers.

We give an introduction to the computational theory of continued fractions, viewed through the lens of matrices and transducers. Then we move to quadratic convergents in terms of the classical rational convergents, which is one of the main topics of the book. With this at hand, the Zopf number and its quadratic convergents are explored through Diophantine analysis. This is followed by the generalized Zopf numbers which can be written compactly in terms of irregular continued fractions, for which many can be shown to have representations by Hurwitz continued fractions. For these Hurwitzian Zopf numbers, we provide an algorithm for converting from irregular to regular continued fractions by using a special type of "interrupted" LR-sequences. Finally, applications to these Hurwitzian Zopf numbers are given, including a refinement of the irrationality measure by iterated logarithms.

Written in an accessible style, the material will be of interest to students and researchers in number theory and approximation theory.

商品描述(中文翻譯)

這本專著源自於對連分數 [1, 2, 3, ...] 的研究，我們稱之為 Zopf 數。其起源可追溯至 1929 年，當時 Siegel 將其引入作為貝塞爾函數的比率。連分數通常以古典風格呈現，且其內容大多透過丟番圖分析來表述。然而，在本書中，計算理論的某些方面可以透過矩陣和轉換器互相使用。

我們將介紹連分數的計算理論，從矩陣和轉換器的角度來看。接著，我們將討論與古典有理收斂相關的二次收斂，這是本書的主要主題之一。在此基礎上，Zopf 數及其二次收斂將透過丟番圖分析進行探討。隨後，我們將介紹可以用不規則連分數緊湊表示的廣義 Zopf 數，許多這些數可以用 Hurwitz 連分數表示。對於這些 Hurwitzian Zopf 數，我們提供了一種算法，通過使用一種特殊類型的「中斷」LR 序列，將不規則連分數轉換為規則連分數。最後，我們將介紹這些 Hurwitzian Zopf 數的應用，包括通過迭代對數的方式來細化無理數度量。

本書以易於理解的風格撰寫，對於數論和近似理論的學生及研究人員將具有興趣。

作者簡介

Carsten Elsner: received Ph.D. from Hannover University in 1990, habilitation from Hannover University in 1997, joined University of Applied Sciences (FHDW) in 2005 as Professor for Mathematics. His research areas are in number theory: continued fractions, Diophantine approximation, transcendental numbers and algebraic independence, recursions, special functions, but also, in universal differential equations. His teaching experience covers the following areas: number theory, approximation theory, combinatorics, mathematics for engineers and computer science students, cryptography, actuarial science, theory of automata, and Petri nets.

Christopher Robin Havens: is the founder of the Prison Mathematics Project (www.prisonmathproject.org), working towards the dissemination and popularization of math to marginalized groups within restrictive environments. His research interests are in the theory of computation and Diophantine analysis in the context of continued fractions.

作者簡介(中文翻譯)

Carsten Elsner：於1990年獲得漢諾威大學的博士學位，1997年獲得漢諾威大學的資格認證，於2005年加入應用科學大學（FHDW）擔任數學教授。他的研究領域包括數論：連分數、丟番圖逼近、超越數和代數獨立性、遞迴、特殊函數，以及普遍微分方程。他的教學經驗涵蓋以下領域：數論、逼近理論、組合數學、工程師和計算機科學學生的數學、密碼學、精算科學、自動機理論和佩特里網。

Christopher Robin Havens：是監獄數學計畫（www.prisonmathproject.org）的創始人，致力於在限制環境中向邊緣群體傳播和普及數學。他的研究興趣集中在計算理論和連分數背景下的丟番圖分析。

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