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In this paper we study the convexity and concavity properties of generalized trigonometric and hyperbolic functions in case of Logarithmic mean.

One place where Belyi functions occur is pullback transformations of hypergeometric differential equations to Fuchsian equations with few singularities. This paper presents a complete classification of Belyi functions for transforming certain hypergeometric equations to Heun equations (i.e., canonical Fuchsian equations with 4 singularities). The considered hypergeometric equations have the local exponent differences 1/k,1/l,1/m that satisfy k,l,m\in N and the hyperbolic condition 1/k+1/l+1/m

Highest-weight representations of infinite dimensional Lie algebras and Hilbert schemes of points are considered, together with the applications of these concepts to partition functions, which are most useful in physics. Partition functions (elliptic genera) are conveniently transformed into product expressions, which may inherit the homology properties of appropriate (poly)graded Lie algebras. Specifically, the role of (Selberg-type) Ruelle spectral functions of hyperbolic geometry in the calculation of partition functions and associated $q$-series are discussed. Examples of these connection in quantum field theory are considered (in particular, within the AdS/CFT correspondence), as the AdS$_{3}$ case where one has Ruelle/Selberg spectral functions, whereas on the CFT side, partition functions and modular forms arise. These objects are here shown to have a common background, expressible in terms of Euler-Poincar\'e and Macdonald identities, which, in turn, describe homological aspects of (finite or infinite) Lie algebra representations. Finally, some other applications of modular forms and spectral functions (mainly related with the congruence subgroup of $SL(2, {\mathbb Z})$) to partition functions, Hilbert schemes of points, and symmetric products are investigated by means of homological and K-theory methods.

We consider the point process of zeroes of certain Gaussian analytic functions and find the asymptotics for the probability that there are more than m points of the process in a fixed disk of radius r, as m-->infinity. For the Planar Gaussian analytic function, sum_n a_n z^n/sqrt(n!), we show that this probability is asymptotic to exp(-0.5 m^2 log(m)). For the Hyperbolic Gaussian analytic functions, sum_n sqrt({-rho choose n}) a_n z^n, rho>0, we show that this probability decays like exp(-cm^2). In the planar case, we also consider the problem posed by Mikhail Sodin on moderate and very large deviations in a disk of radius r as r --> infinity. We partly solve the problem by showing that there is a qualitative change in the asymptotics of the probability as we move from the large deviation regime to the moderate.

Book Source: Digital Library of India Item 2015.201783 dc.contributor.author: Mcmahon, James dc.date.accessioned: 2015-07-09T12:39:35Z dc.date.available: 2015-07-09T12:39:35Z dc.date.digitalpublicationdate: 2005-05-25 dc.date.citation: 1896 dc.identifier.barcode: 2990140054141 dc.identifier.origpath: /data_copy/upload/0054/146 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/201783 dc.description.scannerno: 9 dc.description.scanningcentre: Osmania University dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 87 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Digital Library Of India dc.publisher: John Wiley And Sons dc.rights: Copyright Permitted dc.source.library: Osmania University dc.subject.classification: Natural Sciences dc.subject.classification: Mathematics dc.subject.classification: Number Theory dc.title: Hyperbolic Functions

"A Table and Method of Computation of Electric Wave Propagation, Transmission Line Phenomena, Optical Refraction, and Inverse Hyperbolic Functions of a Complex Variable" is an article from Proceedings of the American Academy of Arts and Sciences, Volume 57 . View more articles from Proceedings of the American Academy of Arts and Sciences . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-20025901

55 p. 21 cm

Book digitized by Google from the library of Harvard University and uploaded to the Internet Archive by user tpb.

Book Source: Digital Library of India Item 2015.164910 dc.contributor.author: George F.becker ,c.e.van Orstrand dc.contributor.other: The Smitthsonian Institution dc.date.accessioned: 2015-07-06T23:36:53Z dc.date.available: 2015-07-06T23:36:53Z dc.date.copyrightexpirydate: 0000-00-00 dc.date.digitalpublicationdate: 2029-09-03 dc.date.citation: 1909 dc.identifier: CMU dc.identifier.barcode: 1990050037168 dc.identifier.origpath: /data/upload/0037/173 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/164910 dc.description.scanningcentre: AKCE, Universal Digital Library dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 374 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Akce-universal Digital Library dc.publisher: The Smitthsonian Institution dc.rights: Out_of_copyright dc.source.library: Cmu dc.subject.classification: Mathematices dc.title: Hyperbolic Functions 1909 dc.rights.holder: The Smitthsonian Institution

We study the power mean inequality of the generalized trigonometric and hyperbolic functions with two parameters. The generalized $p$-trigonometric and $(p, q)$-trigonometric functions were introduced by P. Lindqvist and S. Takeuchi, respectively.

Book Source: Digital Library of India Item 2015.153114 dc.contributor.author: George F. Becker dc.contributor.author: C. E. Van Orstrand dc.date.accessioned: 2015-07-06T19:09:32Z dc.date.available: 2015-07-06T19:09:32Z dc.date.digitalpublicationdate: 2004-05-10 dc.date.citation: 1909 dc.identifier: RMSC, IIIT-H dc.identifier.barcode: 2999990018423 dc.identifier.origpath: /data/upload/0018/428 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/153114 dc.description.scanningcentre: RMSC, IIIT-H dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 386 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Universal Digital Library dc.publisher: The Smithsonian Institution dc.rights: Public Domain dc.title: Hyperbolic Functions

Book digitized by Google from the library of the University of Michigan and uploaded to the Internet Archive by user tpb.

Book digitized by Google from the library of the University of Michigan and uploaded to the Internet Archive by user tpb.

Bibliography: p. 209-212

Bibliography: p. 209-212

Book digitized by Google from the library of the New York Public Library and uploaded to the Internet Archive by user tpb.

Book digitized by Google from the library of the New York Public Library and uploaded to the Internet Archive by user tpb.

Book digitized by Google from the library of the New York Public Library and uploaded to the Internet Archive by user tpb.

Let $M$ be a complete simply connected manifold which is in addition Gromov hyperbolic, coercive and roughly starlike. For a given harmonic function on $M$, a local Fatou Theorem and a pointwise criteria of non-tangential convergence coming from the density of energy are shown: at almost all points of the boundary, the harmonic function converges non-tangentially if and only if the supremum of the density of energy is finite. As an application of these results, a Calder\'on-Stein Theorem is proved, that is, the non-tangential properties of convergence, boundedness and finiteness of energy are equivalent at almost every point of the boundary.

We consider certain double series of Eisenstein type involving hyperbolic-sine functions. We define certain generalized Hurwitz numbers, in terms of which we evaluate those double series. Our main results can be regarded as a certain generalization of well-known results of Hurwitz, Herglotz, Katayama and so on. Our results also include recent formulas of the third-named author which are double analogues of the formulas of Cauchy, Mellin, Ramanujan, Berndt and so on, about certain Dirichlet series involving hyperbolic functions. As an application, we give some evaluation formulas for $q$-zeta functions at positive integers.

We present a rigorous scheme that makes it possible to compute eigenvalues of the Laplace operator on hyperbolic surfaces within a given precision. The method is based on an adaptation of the method of particular solutions to the case of locally symmetric spaces and on explicit estimates for the approximation of eigenfunctions on hyperbolic surfaces by certain basis functions. It can be applied to check whether or not there is an eigenvalue in an \epsilon-neighborhood of a given number \lambda>0. This makes it possible to find all the eigenvalues in a specified interval, up to a given precision with rigorous error estimates. The method converges exponentially fast with the number of basis functions used. Combining the knowledge of the eigenvalues with the Selberg trace formula we are able to compute values and derivatives of the spectral zeta function again with error bounds. As an example we calculate the spectral determinant and the Casimir energy of the Bolza surface and other surfaces.

We present a rigorous scheme that makes it possible to compute eigenvalues of the Laplace operator on hyperbolic surfaces within a given precision. The method is based on an adaptation of the method of particular solutions to the case of locally symmetric spaces and on explicit estimates for the approximation of eigenfunctions on hyperbolic surfaces by certain basis functions. It can be applied to check whether or not there is an eigenvalue in an \epsilon-neighborhood of a given number \lambda>0. This makes it possible to find all the eigenvalues in a specified interval, up to a given precision with rigorous error estimates. The method converges exponentially fast with the number of basis functions used. Combining the knowledge of the eigenvalues with the Selberg trace formula we are able to compute values and derivatives of the spectral zeta function again with error bounds. As an example we calculate the spectral determinant and the Casimir energy of the Bolza surface and other surfaces.

There is a contrast between the two sets of functional equations f_0(x+y) = f_0(x)f_0(y) + f_1(x)f_1(y), f_1(x+y) = f_1(x)f_0(y) + f_0(x)f_1(y), and f_0(x-y) = f_0(x)f_0(y) - f_1(x)f_1(y), f_1(x-y) = f_1(x)f_0(y) - f_0(x)f_1(y) satisfied by the even and odd components of a solution of f(x+y) = f(x) f(y). J. Schwaiger and, later, W. F\"org-Rob and J. Schwaiger considered the extension of these ideas to the case where f is sum of n components. Here we shorten and simplify the statements and proofs of some of these results by a more systematic use of matrix notation.

14

An extension of the finite and infinite Lie groups properties of complex numbers and functions of complex variable is proposed. This extension is performed exploiting hypercomplex number systems that follow the elementary algebra rules. In particular the functions of such systems satisfy a set of partial differential equations that defines an infinite Lie group. Emphasis is put on the functional transformations of a particular two-dimensional hypercomplex number system, capable of maintaining the wave equation as invariant and then the speed of light invariant too. These functional transformations describe accelerated frames and can be considered as a generalization of two dimensional Lorentz group of special relativity. As a first application the relativistic hyperbolic motion is obtained.

Using Jacobi's identity we derive a simple expression for the Bessel functions of integer order in terms of combinations of powers and hyperbolic functions of the same argument.

The goal of this paper is to prove the conjecture of Krzyz posed in 1968 that for nonvanishing holomorphic functions $f(z) = c_0 + c_1 z + ...$ in the unit disk with $|f(z)| \le 1$, we have the sharp bound $|c_n| \le 2/e$ for all $n \ge 1$, with equality only for the function $f(z) = \exp [(z^n - 1)/(z^n + 1)]$ and its rotations. The problem was considered by many researchers, but only partial results have been established. The desired estimate has been proved only for $n \le 5$. Our approach is completely different and relies on complex geometry and pluripotential features of convex domains in complex Banach spaces.

We analyse the role of vertex operator algebra and 2d amplitudes from the point of view of the representation theory of infinite dimensional Lie algebras, MacMahon and Ruelle functions. A p-dimensional MacMahon function is the generating function of p-dimensional partitions of integers. These functions can be represented as amplitudes of a two-dimensional c=1 CFT. In this paper we show that p-dimensional MacMahon functions can be rewritten in terms of Ruelle spectral functions, whose spectrum is encoded in the Patterson-Selberg function of three dimensional hyperbolic geometry.

We obtain new inequalities for certain hypergeometric functions. Using these inequalities, we deduce estimates for the hyperbolic metric and the induced distance function on a certain canonical hyperbolic plane domain.

Singular source terms expressed as weighted summations of Dirac-delta functions are regularized through approximation theory with convolution operators. We consider the numerical solution of scalar and one-dimensional hyperbolic conservation laws with the singular source by spectral Chebyshev collocation methods. The regularization is obtained by convolution with a high-order compactly supported Dirac-delta approximation whose overall accuracy is controlled by the number of vanishing moments, degree of smoothness and length of the support (scaling parameter). An optimal scaling parameter that leads to a high-order accurate representation of the singular source at smooth parts and full convergence order away from the singularities in the spectral solution is derived. The accuracy of the regularization and the spectral solution is assessed by solving an advection and Burgers equation with smooth initial data. Nu- merical results illustrate the enhanced accuracy of the spectral method through the proposed regularization.

In this article, existence results concerning temporal functions with additional properties on a globally hyperbolic manifold are obtained. These properties are certain bounds on geometric quantities as lapse and shift. The results are linked to completeness properties and the existence of closed isometric embeddings in Minkowski spaces.

From the bitsavers.org collection, a scanned-in computer-related document. ibm :: 1620 :: programs :: 7.0.019 Three-In-One Curve Fits Hyperbolic Exponential Power Functions

Book digitized by Google from the library of the University of California and uploaded to the Internet Archive by user tpb.

Book digitized by Google from the library of the University of California and uploaded to the Internet Archive by user tpb.

Book Source: Digital Library of India Item 2015.153131 dc.contributor.author: George F. Becker dc.contributor.author: C. E. Van Orstrand dc.date.accessioned: 2015-07-06T19:09:53Z dc.date.available: 2015-07-06T19:09:53Z dc.date.digitalpublicationdate: 2004-11-05 dc.date.citation: 1931 dc.identifier: RMSC, IIIT-H dc.identifier.barcode: 2999990018439 dc.identifier.origpath: /data/upload/0018/444 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/153131 dc.description.numberedpages: 321 dc.description.numberedpages: 60 dc.description.scanningcentre: RMSC, IIIT-H dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 381 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Universal Digital Library dc.publisher: The Smithsonian Institution dc.rights: Copyright Protected dc.title: Smithsonian Mathematical Tables Hyperbolic Functions

"The Expression of Constant and of Alternating Continued Fractions in Hyperbolic Functions" is an article from The Annals of Mathematics, Volume 9 . View more articles from The Annals of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-1967245

We study weakly hyperbolic iterated function systems on compact spaces, as defined by Edalat, but in the more general setting of a compact parameter space. We prove the existence of attractors, both in the topological and measure theoretical viewpoint, and prove that the measure theoretical attractor is ergodic. We also define weakly hyperbolic iterated functions systems for complete spaces and compact parameter space, and prove that this definition extends the one given by Edalat. Furthermore, we study the question of existence of the attractors in this setting. Finally, we prove a version of the results of Barnsley and Vince about drawing the attractor (also called the chaos game), for the case of compact parameter space.

We study weakly hyperbolic iterated function systems on compact spaces, as defined by Edalat, but in the more general setting of a compact parameter space. We prove the existence of attractors, both in the topological and measure theoretical viewpoint, and prove that the measure theoretical attractor is ergodic. We also define weakly hyperbolic iterated functions systems for complete spaces and compact parameter space, and prove that this definition extends the one given by Edalat. Furthermore, we study the question of existence of the attractors in this setting. Finally, we prove a version of the results of Barnsley and Vince about drawing the attractor (also called the chaos game), for the case of compact parameter space.

Book from Project Gutenberg: Hyperbolic Functions

Book Source: Digital Library of India Item 2015.164911 dc.contributor.author: George F.becker ,c.e.van Orstrand dc.contributor.other: The Smitthsonian Institution dc.date.accessioned: 2015-07-06T23:36:55Z dc.date.available: 2015-07-06T23:36:55Z dc.date.copyrightexpirydate: 0000-00-00 dc.date.digitalpublicationdate: 2029-09-03 dc.date.citation: 1931 dc.identifier: CMU dc.identifier.barcode: 1990050037169 dc.identifier.origpath: /data/upload/0037/174 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/164911 dc.description.scanningcentre: AKCE, Universal Digital Library dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 386 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Akce-universal Digital Library dc.publisher: The Smitthsonian Institution dc.rights: Out_of_copyright dc.source.library: Cmu dc.subject.classification: Mathematices dc.title: Hyperbolic Functions 1931 dc.rights.holder: The Smitthsonian Institution

Chapter 1 is a short history of non-Euclidean geometry, which synthesises my readings of mostly secondary sources. Chapter 2 presents each of the main models of hyperbolic geometry, and describes the tesselation of the upper half-plane induced by the action of $PSL(2,\mathbb{Z})$. Chapter 3 gives background on symmetric spaces and word metrics. Chapter 4 then contains a careful proof of the following theorem of Lubotzky--Mozes--Raghunathan: the word metric on $PSL(2,\mathbb{Z})$ is not Lipschitz equivalent to the metric induced by its action on the associated symmetric space (the upper half-plane), but for $n \geq 3$, these two metrics on $PSL(n,\mathbb{Z})$ are Lipschitz equivalent.

Chapter 1 is a short history of non-Euclidean geometry, which synthesises my readings of mostly secondary sources. Chapter 2 presents each of the main models of hyperbolic geometry, and describes the tesselation of the upper half-plane induced by the action of $PSL(2,\mathbb{Z})$. Chapter 3 gives background on symmetric spaces and word metrics. Chapter 4 then contains a careful proof of the following theorem of Lubotzky--Mozes--Raghunathan: the word metric on $PSL(2,\mathbb{Z})$ is not Lipschitz equivalent to the metric induced by its action on the associated symmetric space (the upper half-plane), but for $n \geq 3$, these two metrics on $PSL(n,\mathbb{Z})$ are Lipschitz equivalent.

Chapter 1 is a short history of non-Euclidean geometry, which synthesises my readings of mostly secondary sources. Chapter 2 presents each of the main models of hyperbolic geometry, and describes the tesselation of the upper half-plane induced by the action of $PSL(2,\mathbb{Z})$. Chapter 3 gives background on symmetric spaces and word metrics. Chapter 4 then contains a careful proof of the following theorem of Lubotzky--Mozes--Raghunathan: the word metric on $PSL(2,\mathbb{Z})$ is not Lipschitz equivalent to the metric induced by its action on the associated symmetric space (the upper half-plane), but for $n \geq 3$, these two metrics on $PSL(n,\mathbb{Z})$ are Lipschitz equivalent.

Book digitized by Google from the library of Harvard University and uploaded to the Internet Archive by user tpb.

1 p. \U+25a1\., 77 p. 24 cm

Book Source: Digital Library of India Item 2015.153153 dc.contributor.author: A. E. Kennelly dc.date.accessioned: 2015-07-06T19:10:19Z dc.date.available: 2015-07-06T19:10:19Z dc.date.digitalpublicationdate: 2004-11-05 dc.date.citation: 1916 dc.identifier: RMSC, IIIT-H dc.identifier.barcode: 2999990018459 dc.identifier.origpath: /data/upload/0018/464 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/153153 dc.description.numberedpages: 302 dc.description.numberedpages: 20 dc.description.scanningcentre: RMSC, IIIT-H dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 322 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Universal Digital Library dc.publisher: Mcgraw-hill Book Company dc.rights: Public Domain dc.title: The Application Of Hyperbolic Functions To Electrical Engineering Problems

The article offers an alternative version of the system of independent functions. New functions mutually simultaneously are expressed through the usual trigonometric and hyperbolic functions and are therefore called trigohyperbolic functions. Obtained relations between new functions, as well as between new and traditional ones functions. The unusualness of these ratios makes the theory quite complex, but promising.

Inverse Hyperbolic Trigonometric Functions • Language: Syriac • Author: Ashur Cherry (Toronto, Canada)

Derivatives and primitives for all hyperbolic functions.