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Sunday, July 12, 2020 | History

3 edition of Hyperbolic Fibonacci and Lucas functions found in the catalog.

Hyperbolic Fibonacci and Lucas functions

A. P. Stakhov

# Hyperbolic Fibonacci and Lucas functions

## by A. P. Stakhov

Written in English

Subjects:
• Fibonacci numbers.,
• Lucas numbers.,
• Exponential functions.

• Edition Notes

Bibliography: p. 236-237.

The Physical Object ID Numbers Statement by Alexey Stakhov. Pagination 237 p. : Number of Pages 237 Open Library OL20297340M ISBN 10 966843207X OCLC/WorldCa 225646372

The Fibonacci and Lucas numbers and are entire analytical functions of that are defined over the whole complex -plane: Periodicity. The Fibonacci and Lucas numbers and do not have periodicity. Parity and symmetry. The Fibonacci and Lucas numbers and generically do . The representation of Fibonacci and Lucas numbers in terms of hyperbolic functions [9, p. 7 ff.] and the idea of deriving Fibonacci identities from known hyperbolic identities are not new (e.g., see [6]).Author: Piero Filipponi, Herta T. Freitag.

Recounting Fibonacci and Lucas Identities Arthur T. Benjamin Harvey Mudd College Jennifer J. Quinn book series for MAA, and an Associate Editor of Mathematics functions, determinants, hyperbolic functions, and so on) [10]. While these tech? Hyperbolic Fibonacci and Lucas functions, aka a new maths article that's too "essay-like" for creation at the moment Hullo mathematical peoples. A new editor has written Wikipedia talk:Articles for creation/Hyperbolic Fibonacci and Lucas functions, which looks very well referenced (I've pointed him to how to do inline citations properly).

The hyperbolic functions are defined in terms of the natural exponential function ex. For example, the hyperbolic sine. function is defined as (ex – e–x)/2 and denoted sinh, pronounced “ shin ”, so that sinh x = (ex – e–x)/2. #N#We'll see later on the reasons why these functions are named the way they are. There are 6 hyperbolic. The book is intended for a wide audience including mathematics teachers of high schools, students of colleges and universities and scientists in the field of mathematics, theoretical physics and computer science. Mathematics of Harmony: Generalizations of Fibonacci Numbers and the Golden Mean; Hyperbolic Fibonacci and Lucas Functions.

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### Hyperbolic Fibonacci and Lucas functions by A. P. Stakhov Download PDF EPUB FB2

Hyperbolic functions—hyperbolic Fibonacci and Lucas functions. Ukrainian mathematician academician Yury Mitropolsky, Hyperbolic Fibonacci and Lucas functions book article on the hyperbolic Fibonacci and Lucas func-tions was published by the Ukrainian mathematicians Alexey Stakhov and Ivan Tkachenko in the Reports of the National Academy of Sciences of Ukraine in [6].File Size: KB.

functions [4, 5] called in [4] hyperbolic Fibonacci and Lucas λ-functions. They are a wide generalization of the classical hyperbolic functions and hyperbolic Fibonacci and Lucas functions introduced in [2, 3]. The main goal of this article is to state a general theory of hyperbolic functions,File Size: KB.

PDF | This article refers to the “Mathematics of Harmony” by Alexey Stakhov ina new interdisciplinary direction of modern science.

The main goal | Find, read and cite all the. This single-volume compilation of three books centers on Hyperbolic Functions, an introduction to the relationship between the hyperbolic sine, cosine, and tangent, and the geometric properties of the hyperbola.

The development of the hyperbolic functions, in addition to those of the trigonometric (circular) functions, appears in parallel columns for comparison.

Hyperbolic Fibonacci and Lucas Functions (Stakhov-Tkachenko's Definition) A Definition of Hyperbolic Fibonacci and Lucas Functions.

The Hyperbolic Fibonacci and Lucas Tangent and Cotangent. Some Properties of the Hyperbolic Fibonacci and Lucas Functions. Hyperbolic Fibonacci Sine. Hyperbolic Fibonacci Cosine. Hyperbolic Lucas Sine. Hyperbolic. and are called Fibonacci hyperbolic functions.

Now, equalities introduced [12] can naturally be related with the q-Fibonacci hyperbolic functions studied before. The following correlations that are similar to the equation [cosh(x)]2 − [sinh(x)]2 = 1 are valid for the Fibonacci hyperbolic functions: [cosFh(x)]2 −[sinFh(x)]2 = 4 5.

(2)Author: A. Güncan, Y. Erbil. Based on the definitions () and (), Stakhov [17] and later Falcon and Plaza [23] introduced an important class of the elementary functions called the Hyperbolic Fibonacci and Lucas Functions.

Hyperbolic Fibonacci and Lucas Functions, “Golden” Fibonacci Goniometry, Bodnar’s Geometry, and Hilbert’s Fourth Problem —Part II. A New Geometric Theory of Phyllotaxis (Bodnar’s Geometry) Alexey Stakhov1,2, Samuil Aranson3 1International Higher Education Academy of Sciences, Moscow Russia.

Assisted by Scott Olsen (Central Florida Community College, USA). This volume is a result of the author's four decades of research in the field of Fibonacci numbers and the Golden Section and their applications.

It provides a broad introduction to the fascinating and beautiful subject of the OC Mathematics of Harmony, OCO a new interdisciplinary direction of modern science.5/5(1). The hyperbolic functions take a real argument called a hyperbolic size of a hyperbolic angle is twice the area of its hyperbolic hyperbolic functions may be defined in terms of the legs of a right triangle covering this sector.

In complex analysis, the hyperbolic functions arise as the imaginary parts of sine and hyperbolic sine and the hyperbolic cosine are. Recently in the works of Alexey Stakhov and Boris Rozin the classical results in the Fibonacci numbers theory were generalized [].

Consider some of these generalizations. Hyperbolic Fibonacci and Lucas functions Stakhov and Rozin introduced in [14] the so. The Golden Ratio, Fibonacci Numbers, and the "Golden" Hyperbolic Fibonacci and Lucas Functions; It describes a general theory of "recursive" hyperbolic functions based on the "golden," "silver," and other "metallic" proportions.

The book contains an original solution of Hilbert's Fourth Problem for hyperbolic and spherical geometries, and Author: Alexey Stakhov, Samuil Aranson, Scott Olsen. An extension of the classical hyperbolic functions is introduced and studied. These new k-Fibonacci hyperbolic functions generalize also the k-Fibonacci sequences, say {F k, n} n = 0 ∞, recently found by studying the recursive application of two geometrical transformations onto C ¯ = C ∪ {+ ∞} used in the well-known four-triangle longest-edge (4TLE) by: Hyperbolic functions sinh and cosh The hyperbolic functions sinh (pronounced “shine”) and cosh are deﬁned by the formulae coshx = ex +e−x 2 sinhx = ex −e−x 2 (1) The function coshx is an even function, and sinhx is odd.

On modern calculators hyperbolic functions are File Size: 81KB. Based on an analogy between Binet's formulas, and the classical hyperbolic functions: sh(x)= e x − e −x 2; ch(x)= e x + e −x 2, we can give the following definitions of the hyperbolic Fibonacci and Lucas functions that are different from the definitions,: Symmetrical Fibonacci sine (8) sFs(x)= α x −α −x 5.

Symmetrical Fibonacci cosine (9) cFs(x)= α x +α −x by: Hyperbolic Functions has been added to your Cart Add to Cart. Buy Now See all 65 formats and editions Hide other formats and editions. Price New from Used from Kindle "Please retry" $— 5/5(2). bonacci and Lucas numbers. The results have a general character and contain some of the existing results as special cases. Mathematics Subject Classi cation: 11B39, 11Y60 Keywords: Fibonacci number, Lucas number, Hyperbolic inverse tangent, Product identity. 1 Introduction The Fibonacci numbers F n and Lucas numbers L n are de ned for n 0 as, F Author: Robert Frontczak. The hyperbolic functions coshx and sinhx are deﬁned using the exponential function ex. We shall start with coshx. This is deﬁned by the formula coshx = ex +e−x 2. We can use our knowledge of the graphs of ex and e−x to sketch the graph of coshx. First, let us calculate the value of. Lobachevsky's hyperbolic geometry is based on classical hyperbolic functions. Replacing the classical ones by the hyperbolic$\lambda\$-Fibonacci functions they get different hyperbolic geometries. To obtain a similar generalization for spherical geometry, yet another type of Fibonacci functions are needed.

Hyperbolic functions CRTM, Thus trig identities can be directly related to hyperbolic identities, except that whenever sin2 x ap- If you wish for more detail on any of this you should consult an A-level book such as Bostock & Chandler’s Further Pure Mathematics.

Thus, the introduction of the hyperbolic Fibonacci and Lucas functions is raising “Fibonacci numbers theory” [32] on a much higher scientific level. Now, we will discuss a “physical” sense of the hyperbolic Fibonacci and Lucas functions.

A brilliant answer to this question is .the golden non euclidean geometry Download the golden non euclidean geometry or read online books in PDF, EPUB, Tuebl, and Mobi Format. Click Download or Read Online button to get the golden non euclidean geometry book now.

This site is like a library, Use search box in .The hyperbolic trigonometric functions extend the notion of the parametric equations for a unit circle (x = cos ⁡ t (x = \cos t (x = cos t and y = sin ⁡ t) y = \sin t) y = sin t) to the parametric equations for a hyperbola, which yield the following two fundamental hyperbolic equations.

x = cosh ⁡ a = e a + e − a 2, y = sinh ⁡ a = e a − e − a 2. x = \cosh a = \dfrac{e^a + e^{-a.