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Showing 1 to 20 of 33 for “"continued fractions"”.
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Approximations by means of continued fractions
… when extended, leads to our modern notion of continued fractions. The classical theory of continued fractions began during the Renaissance when Arabic numerals and the modern fractional notation had become common. It was studied and extended until about the end of the nineteenth century. This …
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On the convergence of continued fractions
… paper, fundamental definitions in the theory of continued fractions are set forth. In Chapter II we give certain classical convergence theorems for continued fractions of various types. The results of Chapter III are original and sharpen a Mom convergence theorem due to Leighton and Wall.
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Continued fractions and representations of graphs
Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2020-08-25 without embargo terms
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Q-Continued Fractions and Related Q-Series
… successive approximation. Also, we prove several continued fractions of Ramanujan which are related to the odd parts of continued fractions.
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Some important continued fractions of Ramanujan and Selberg
… which arise in Ramanujan and Selberg's work on q-continued fractions. From these solutions, we derive criteria for the convergence of three Ramanujan-Selberg continued fractions when q is a primitive m-th root of unity. Moreover, when the continued fractions converge, we determine their values …
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Advances in the theory of p-adic continued fractions
L'abstract è presente nell'allegato / the abstract is in the attachment
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Applications of continued fractions in one and more variables
Elementary properties of continued fractions are derived from sets of three-term recurrence relations and approximation methods are developed from this simple approach. First, a well-known method for numerical inversion of Laplace transforms is modified in two different ways to obtain exponential …
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Rates of Convergence of Continued Fractions and an Approximation Theorem
Embargo set by: Seth Robbins for item 88106 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs
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On the Rogers-Ramanujan and Ramanujan-Gollnitz-Gordon Continued Fractions
In a manuscript of Ramanujan, published with his Lost Notebook, there are forty identities involving the Rogers-Ramanujan functions. According to G. N. Watson, the beauty of these identities is comparable to that of the Rogers-Ramanujan identities. In the final chapter, we establish modular …
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Geometric and ergodic properties of certain classes of continued fractions
Made available in DSpace on 2019-11-26T20:33:47Z (GMT). No. of bitstreams: 2 MERRIMAN-DISSERTATION-2019.pdf: 3545539 bytes, checksum: 1ceb6dccbf49a654427e8d1f15d2be3d (MD5) LICENSE.txt: 4212 bytes, checksum: b6441bf949f57dfb6f36245b79c18128 (MD5) Previous issue date: 2019-07-08
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Chebyshev-like polynomials, conic distribution of roots, and continued fractions
… of Chebyshev polynomials. Here we use continued fractions to give estimates for the roots that do not lie in the interval $(-1,1)$. We then show the connection between polynomials with roots on concentric circles to polynomials with roots on ellipses. In particular, we construct a …
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Applications of dynamical systems to Farey sequences and continued fractions
… SL(2,R) to study the spacing statistics of Farey fractions. For a given finite index subgroup H ⊆ SL(2,Z), we use a process developed by Fisher and Schmidt to lift a cross section of the horocycle flow on SL(2,R)/SL(2,Z) found by Athreya and Cheung to the finite cover SL(2,R)/H of SL(2,R)/SL(2,Z). …
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Hypergeometric functions, continued fractions for products of gamma functions, and q-analogues
Some of the most interesting of Ramanujan's continued fraction identities are those involving ratios of Gamma functions in Chapter 12 of his second notebook. This thesis develops a method for deriving such identities, using hypergeometric functions as the main tool. We begin by deriving a continued …
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Contributions to Ramanujan's continued fractions, class invariants, partition identities and modular equations
Various topics related to the work of Ramanujan are discussed in this thesis. In Chapter 2, we give a new proof of Ramanujan's famous partition identity modulo 5 (see (1.1)). This proof is an improvement of W. N. Bailey's proof given in 1952. We also establish a new proof of Ramanujan's partition …
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On the Convergence and Divergence of Q-Continued Fractions on and Off the Unit Circle
… implies general convergence. We show that all continued fractions in a certain class, which includes the Rogers-Ramanujan continued fractions and the three ""Ramanujan-Selberg"" continued fractions, diverge in the general sense at an uncountable set of points on the unit circle. We also show …
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On the application of the convergence theorems of continued fractions to the construction of electrical networks
Thesis: M.S., Massachusetts Institute of Technology, Department of Mathematics, 1934
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An investigation of the use of inertia as a perturbation parameter and continued fractions in linear vibration problems
… 2) The expansion of these series solutions as continued fractions. The series solutions are obtained by applying the classical perturbation technique, which assumes the solution for the governing differential equation (say, for the case of one independent and one dependent variable) can be …
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