The Hyperbolic Cosine Cosh(x) and Sine Sinh(x) Functions

2008 ◽  
pp. 269-279
Author(s):  
Keith B. Oldham ◽  
Jan C. Myland ◽  
Jerome Spanier
Keyword(s):  
Hyperbolic Cosine

scholarly journals Approaching the self-dual point of the sinh-Gordon model

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Robert Konik ◽  
Márton Lájer ◽  
Giuseppe Mussardo
Keyword(s):  
Small Volume ◽  
Zero Mode ◽  
Form Factors ◽  
Lagrangian Formulation ◽  
The Self ◽  
Quantum Mechanical ◽  
Exact Form ◽  
Coupling Constant ◽  
Self Duality ◽  
Hyperbolic Cosine

Abstract One of the most striking but mysterious properties of the sinh-Gordon model (ShG) is the b → 1/b self-duality of its S-matrix, of which there is no trace in its Lagrangian formulation. Here b is the coupling appearing in the model’s eponymous hyperbolic cosine present in its Lagrangian, cosh(bϕ). In this paper we develop truncated spectrum methods (TSMs) for studying the sinh-Gordon model at a finite volume as we vary the coupling constant. We obtain the expected results for b ≪ 1 and intermediate values of b, but as the self-dual point b = 1 is approached, the basic application of the TSM to the ShG breaks down. We find that the TSM gives results with a strong cutoff Ec dependence, which disappears according only to a very slow power law in Ec. Standard renormalization group strategies — whether they be numerical or analytic — also fail to improve upon matters here. We thus explore three strategies to address the basic limitations of the TSM in the vicinity of b = 1. In the first, we focus on the small-volume spectrum. We attempt to understand how much of the physics of the ShG is encoded in the zero mode part of its Hamiltonian, in essence how ‘quantum mechanical’ vs ‘quantum field theoretic’ the problem is. In the second, we identify the divergencies present in perturbation theory and perform their resummation using a supra-Borel approximate. In the third approach, we use the exact form factors of the model to treat the ShG at one value of b as a perturbation of a ShG at a different coupling. In the light of this work, we argue that the strong coupling phase b > 1 of the Lagrangian formulation of model may be different from what is naïvely inferred from its S-matrix. In particular, we present an argument that the theory is massless for b > 1.

Electroosmotic Flow of Power-Law Fluids in a Slit Microchannel

2009 ◽  
Author(s):  
Cunlu Zhao ◽  
Chun Yang
Keyword(s):  
Shear Stress ◽  
Double Layer ◽  
Power Law ◽  
Dynamic Viscosity ◽  
Electroosmotic Flow ◽  
Flow Behavior ◽  
Flow Behavior Index ◽  
Power Law Fluids ◽  
Hyperbolic Cosine ◽  
Behavior Index

Electroosmotic flow of power-law fluids in a slit channel is analyzed. The governing equations including the linearized Poisson–Boltzmann equation, the Cauchy momentum equation and the continuity equation are solved to seek analytical expressions for the shear stress, dynamic viscosity and velocity distributions. Specifically, exact solutions of the velocity distributions are explicitly found for several special values of the flow behavior index. Furthermore, with the implementation of an approximate scheme for the hyperbolic cosine function, approximate solutions of the velocity distributions are obtained. In addition, a mathematical expression for the average electroosmotic velocity is derived for large values of the dimensionless electrokinetic parameter, κH, in a fashion similar to the Smoluchowski equation. Hence, a generalized Smoluchowski velocity is introduced by taking into account contributions due to the finite thickness of the electric double layer and the flow behavior index of power-law fluids. Finally, calculations are performed to examine the effects of κH, flow behavior index, double layer thickness, and applied electric field on the shear stress, dynamic viscosity, velocity distribution, and average velocity/flow rate of the electroosmotic flow of power-law fluids.

scholarly journals Tachyonic matter cosmology with exponential and hyperbolic potentials

2017 ◽  
Vol 26 (02) ◽  
pp. 1750012 ◽  
Author(s):  
B. Pourhassan ◽  
J. Naji
Keyword(s):  
Scalar Field ◽  
Observational Data ◽  
Cosmological Parameters ◽  
Field Potentials ◽  
Exponential Potential ◽  
Frw Universe ◽  
Hyperbolic Cosine ◽  
Special Cases ◽  
Friedmann Robertson Walker ◽  
Type Potential

In this paper, we consider tachyonic matter in spatially flat Friedmann–Robertson–Walker (FRW) universe, and obtain behavior of some important cosmological parameters for two special cases of potentials. First, we assume the exponential potential and then consider hyperbolic cosine type potential. In both cases, we obtain behavior of the Hubble, deceleration and EoS parameters. Comparison with observational data suggest the model with hyperbolic cosine type scalar field potentials has good model to describe universe.

Sign in / Sign up

Export Citation Format

Share Document