Continued Fractions A Modern and Classical Journey into the World of Siegel's Continued Fractions /
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...
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| Format: | Électronique eBook |
| Langue: | anglais |
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Springer Nature Switzerland : Imprint: Springer,
2025.
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| Édition: | 1st ed. 2025. |
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| Accès en ligne: | https://doi.org/10.1007/978-3-031-99546-0 |
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| 100 | 1 | |a Elsner, Carsten. |e author. |4 aut |4 http://id.loc.gov/vocabulary/relators/aut | |
| 245 | 1 | 0 | |a Continued Fractions |h [electronic resource] : |b A Modern and Classical Journey into the World of Siegel's Continued Fractions / |c by Carsten Elsner, Christopher Robin Havens. |
| 250 | |a 1st ed. 2025. | ||
| 264 | 1 | |a Cham : |b Springer Nature Switzerland : |b Imprint: Springer, |c 2025. | |
| 300 | |a 1 online resource (XVII, 366 p. 14 illus.). |b 13 illus. in color. | ||
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| 505 | 0 | |a I. A classical introduction to continued fractions and quadratic convergents -- II. The Zopf constant [1,2,3,...] and its relatives -- III. Matrices and transducers: The computational theory of continued fractions -- IV. The theory of conversions from irregular to regular Hurwitz continued fractions -- V. On a refinement of the irrationality measure -- VI. Appendix. | |
| 520 | |a 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 it's 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. | ||
| 532 | 8 | |a Accessibility summary: This PDF has been created in accordance with the PDF/UA-1 standard to enhance accessibility, including screen reader support, described non-text content (images, graphs), bookmarks for easy navigation, keyboard-friendly links and forms and searchable, selectable text. We recognize the importance of accessibility, and we welcome queries about accessibility for any of our products. If you have a question or an access need, please get in touch with us at accessibilitysupport@springernature.com. Please note that a more accessible version of this eBook is available as ePub. | |
| 532 | 8 | |a No reading system accessibility options actively disabled | |
| 532 | 8 | |a Publisher contact for further accessibility information: accessibilitysupport@springernature.com | |
| 588 | |a Print version record. | ||
| 650 | 0 | |a Number theory. | |
| 650 | 1 | 4 | |a Number Theory. |
| 650 | 2 | 4 | |a Computational Number Theory. |
| 700 | 1 | |a Havens, Christopher Robin. |e author. |4 aut |4 http://id.loc.gov/vocabulary/relators/aut | |
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