Highly accurate tables for elementary functions

1995 ◽  
Vol 35 (3) ◽  
pp. 352-360
Author(s):  
Wolfram Luther
Keyword(s):  
Elementary Functions

scholarly journals Second-order integrable Lagrangians and WDVV equations

2021 ◽  
Vol 111 (2) ◽  
Author(s):  
E. V. Ferapontov ◽  
M. V. Pavlov ◽  
Lingling Xue
Keyword(s):  
Lagrangian Density ◽  
Theta Functions ◽  
Second Order ◽  
Elementary Functions ◽  
Lagrange Equations ◽  
Wdvv Equations ◽  
Integrability Conditions ◽  
Jacobi Theta Functions

AbstractWe investigate the integrability of Euler–Lagrange equations associated with 2D second-order Lagrangians of the form $$\begin{aligned} \int f(u_{xx},u_{xy},u_{yy})\ \mathrm{d}x\mathrm{d}y. \end{aligned}$$ ∫ f ( u xx , u xy , u yy ) d x d y . By deriving integrability conditions for the Lagrangian density f, examples of integrable Lagrangians expressible via elementary functions, Jacobi theta functions and dilogarithms are constructed. A link of second-order integrable Lagrangians to WDVV equations is established. Generalisations to 3D second-order integrable Lagrangians are also discussed.

scholarly journals Quantum periods and TBA-like equations for a class of Calabi-Yau geometries

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Bao-ning Du ◽  
Min-xin Huang
Keyword(s):  
Large Class ◽  
Bethe Ansatz ◽  
Difference Equations ◽  
The Other ◽  
Elementary Functions ◽  
Quantum Dilogarithm ◽  
Thermodynamic Bethe Ansatz ◽  
Dilogarithm Function ◽  
Quantum Operators

Abstract We continue the study of a novel relation between quantum periods and TBA(Thermodynamic Bethe Ansatz)-like difference equations, generalize previous works to a large class of Calabi-Yau geometries described by three-term quantum operators. We give two methods to derive the TBA-like equations. One method uses only elementary functions while the other method uses Faddeev’s quantum dilogarithm function. The two approaches provide different realizations of TBA-like equations which are nevertheless related to the same quantum period.

scholarly journals Application of the Efros Theorem to the Function Represented by the Inverse Laplace Transform of s−μ exp(−sν)

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 354
Author(s):  
Alexander Apelblat ◽  
Francesco Mainardi
Keyword(s):  
Laplace Transform ◽  
Operational Calculus ◽  
Inverse Laplace Transform ◽  
Elementary Functions ◽  
Hyperbolic Functions ◽  
Error Functions ◽  
Convolution Integrals ◽  
Special Case ◽  
Integral Identities ◽  
Finite Integrals

Using a special case of the Efros theorem which was derived by Wlodarski, and operational calculus, it was possible to derive many infinite integrals, finite integrals and integral identities for the function represented by the inverse Laplace transform. The integral identities are mainly in terms of convolution integrals with the Mittag–Leffler and Volterra functions. The integrands of determined integrals include elementary functions (power, exponential, logarithmic, trigonometric and hyperbolic functions) and the error functions, the Mittag–Leffler functions and the Volterra functions. Some properties of the inverse Laplace transform of s−μexp(−sν) with μ≥0 and 0<ν<1 are presented.

Introduction to the Elementary Functions

1971 ◽  
Vol 55 (391) ◽  
pp. 103
Author(s):  
G. Merlane ◽  
R. A. Knight ◽  
W. E. Hoff
Keyword(s):  
Elementary Functions

A study of plane wave and fundamental solution in the theory of microstretch thermoelastic diffusion solid with phase-lag models

2015 ◽  
Vol 11 (2) ◽  
pp. 160-185 ◽  
Author(s):  
Rajneesh Kumar ◽  
Sanjeev Ahuja ◽  
S.K. Garg
Keyword(s):  
Fundamental Solution ◽  
Plane Wave ◽  
Phase Lag ◽  
Dual Phase ◽  
Elementary Functions ◽  
Thermoelastic Diffusion ◽  
Content Type ◽  
Propagation Technique ◽  
Steady Oscillations ◽  
Phase Velocity And Attenuation

Purpose – The purpose of this paper is to study of propagation of plane wave and the fundamental solution of the system of differential equations in the theory of a microstretch thermoelastic diffusion medium in phase-lag models for the case of steady oscillations in terms of elementary functions. Design/methodology/approach – Wave propagation technique along with the numerical methods for computation using MATLAB software has been applied to investigate the problem. Findings – Characteristics of waves like phase velocity and attenuation coefficient are computed numerically and depicted graphically. It is found that due to the presence of diffusion effect, these characteristics get influenced significantly. However, due to decoupling of CD-I and CD-II waves from rest of other, no effect on these characteristics can be perceived. Originality/value – Basic properties of the fundamental solution are established by introducing the dual-phase-lag diffusion (DPLD) and dual-phase-lag heat transfer (DPLT) models.

THE TWISTED EUCLIDEAN GREEN’S FUNCTION IN THE SPACETIME OF A COSMIC STRING

1992 ◽  
Vol 01 (02) ◽  
pp. 371-377 ◽  
Author(s):  
B. LINET
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Minkowski Spacetime ◽  
Vacuum Expectation ◽  
Vacuum Expectation Value ◽  
Massive Scalar ◽  
Elementary Functions ◽  
Integral Expression ◽  
Massive Scalar Field ◽  
Expectation Value

In a conical spacetime, we determine the twisted Euclidean Green’s function for a massive scalar field. In particular, we give a convenient form for studying the vacuum averages. We then derive an integral expression of the vacuum expectation value <Φ2(x)>. In the Minkowski spacetime, we express <Φ2(x)> in terms of elementary functions.

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