scholarly journals Polynomial-Exponential Bounds for Some Trigonometric and Hyperbolic Functions

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 308
Author(s):  
Yogesh J. Bagul ◽  
Ramkrishna M. Dhaigude ◽  
Marko Kostić ◽  
Christophe Chesneau
Keyword(s):  
Exponential Type ◽  
Bernoulli Numbers ◽  
Cosine Function ◽  
Sinc Function ◽  
Series Expansions ◽  
Trigonometric Functions ◽  
Hyperbolic Functions ◽  
Double Inequality ◽  
Exponential Bounds ◽  
Cosine Functions

Recent advances in mathematical inequalities suggest that bounds of polynomial-exponential-type are appropriate for evaluating key trigonometric functions. In this paper, we innovate in this sense by establishing new and sharp bounds of the form (1−αx2)eβx2 for the trigonometric sinc and cosine functions. Our main result for the sinc function is a double inequality holding on the interval (0, π), while our main result for the cosine function is a double inequality holding on the interval (0, π/2). Comparable sharp results for hyperbolic functions are also obtained. The proofs are based on series expansions, inequalities on the Bernoulli numbers, and the monotone form of the l’Hospital rule. Some comparable bounds of the literature are improved. Examples of application via integral techniques are given.

scholarly journals Relation of Some Known Functions in terms of Generalized Meijer G -Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Syed Ali Haider Shah ◽  
Shahid Mubeen ◽  
Gauhar Rahman ◽  
Jihad Younis
Keyword(s):  
Bessel Functions ◽  
Logarithmic Function ◽  
Trigonometric Functions ◽  
Modified Bessel Functions ◽  
Modified Bessel Function ◽  
Exponential Functions ◽  
Hyperbolic Functions ◽  
Binomial Expansion ◽  
Sine And Cosine Functions ◽  
Cosine Functions

The aim of this paper is to prove some identities in the form of generalized Meijer G -function. We prove the relation of some known functions such as exponential functions, sine and cosine functions, product of exponential and trigonometric functions, product of exponential and hyperbolic functions, binomial expansion, logarithmic function, and sine integral, with the generalized Meijer G -function. We also prove the product of modified Bessel function of first and second kind in the form of generalized Meijer G -function and solve an integral involving the product of modified Bessel functions.

scholarly journals Integrated cosine functions

1996 ◽  
Vol 19 (3) ◽  
pp. 575-580 ◽  
Author(s):  
Quan Zheng
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Second Order ◽  
Basic Theory ◽  
Cosine Function ◽  
Integrated Semigroups ◽  
Approximation Theorems ◽  
The Relationship ◽  
Cosine Functions

In order to the second order Cauchy problem(CP2):x″(t)=Ax(t),x(0)=x∈D(An),x″(0)=y∈D(Am)on a Banach space, Arendt and Kellermann recently introduced the integrated cosine function. This paper is concerned with its basic theory, which contain some properties, perturbation and approximation theorems, the relationship to analytic integrated semigroups, interpolation and extrapolation theorems.

scholarly journals Hypercyclic and chaotic integrated C-cosine functions

Filomat ◽  
2012 ◽  
Vol 26 (1) ◽  
pp. 1-44 ◽  
Author(s):  
Marko Kostic
Keyword(s):  
Structural Properties ◽  
Theoretical Approach ◽  
Cosine Function ◽  
Cosine Functions

The main purpose of the paper is to display the main structural properties of hypercyclic and chaotic integrated C-cosine functions. The notions of hypercyclicity, mixing and chaoticity of an ?-times integrated C-cosine function (??0) are defined by using distributional techniques. We provide several examples which justify our abstract theoretical approach.

scholarly journals p-Trigonometric andp-Hyperbolic Functions in Complex Domain

2016 ◽  
Vol 2016 ◽  
pp. 1-18
Author(s):  
Petr Girg ◽  
Lukáš Kotrla
Keyword(s):  
Differential Equations ◽  
Complex Domain ◽  
Trigonometric Functions ◽  
Maclaurin Series ◽  
Hyperbolic Functions

We study extension ofp-trigonometric functionssinpandcospand ofp-hyperbolic functionssinhpandcoshpto complex domain. Our aim is to answer the question under what conditions onpthese functions satisfy well-known relations for usual trigonometric and hyperbolic functions, such as, for example,sin(z)=-i·sinh⁡i·z. In particular, we prove in the paper that forp=6,10,14,…thep-trigonometric andp-hyperbolic functions satisfy very analogous relations as their classical counterparts. Our methods are based on the theory of differential equations in the complex domain using the Maclaurin series forp-trigonometric andp-hyperbolic functions.

scholarly journals A Sharp Double Inequality for Trigonometric Functions and Its Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Zhen-Hang Yang ◽  
Yu-Ming Chu ◽  
Ying-Qing Song ◽  
Yong-Min Li
Keyword(s):  
Trigonometric Functions ◽  
Double Inequality

We present the best possible parameterspandqsuch that the double inequality(2/3)cos2p(t/2)+1/31/p<sin t/t<(2/3)cos2q(t/2)+1/31/qholds for anyt∈(0,π/2). As applications, some new analytic inequalities are established.

Complex Wave Solutions to Mathematical Biology Models I: Newell–Whitehead–Segel and Zeldovich Equations

2018 ◽  
Vol 13 (8) ◽  
Author(s):  
Alper Korkmaz
Keyword(s):  
Gordon Equation ◽  
Reaction Diffusion ◽  
Expansion Method ◽  
Reaction Diffusion Equations ◽  
Trigonometric Functions ◽  
Sine Gordon Equation ◽  
Hyperbolic Functions ◽  
Complex Wave ◽  
Complex Valued ◽  
Homogeneous Balance

Complex and real valued exact solutions to some reaction-diffusion equations are suggested by using homogeneous balance and Sine-Gordon equation expansion method. The predicted solution of finite series of some hyperbolic functions is determined by using some relations between the hyperbolic functions and the trigonometric functions based on Sine-Gordon equation and traveling wave transform. The Newel–Whitehead–Segel (NWSE) and Zeldovich equations (ZE) are solved explicitly. Some complex valued solutions are depicted in real and imaginary components for some particular choice of parameters.

An Algorithm for the Inverse Solution of Geodesic Sailing without Auxiliary Sphere

2014 ◽  
Vol 67 (5) ◽  
pp. 825-844 ◽  
Author(s):  
Wei-Kuo Tseng
Keyword(s):  
Inverse Problem ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Computer Algebra ◽  
Computer Algebra System ◽  
Computational Cost ◽  
Inverse Solution ◽  
Series Expansions ◽  
Trigonometric Functions ◽  
Innovative Method

An innovative algorithm to determine the inverse solution of a geodesic with the vertex or Clairaut constant located between two points on a spheroid is presented. This solution to the inverse problem will be useful for solving problems in navigation as well as geodesy. The algorithm to be described derives from a series expansion that replaces integrals for distance and longitude, while avoiding reliance on trigonometric functions. In addition, these series expansions are economical in terms of computational cost. For end points located at each side of a vertex, certain numerical difficulties arise. A finite difference method together with an innovative method of iteration that approximates Newton's method is presented which overcomes these shortcomings encountered for nearly antipodal regions. The method provided here, which does not involve an auxiliary sphere, was aided by the Computer Algebra System (CAS) that can yield arbitrarily truncated series suitable to the users accuracy objectives and which are limited only by machine precisions.

scholarly journals The Improved Generalized tanh-coth Method Applied to Sixth-Order Solitary Wave Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-11
Author(s):  
M. Torvattanabun ◽  
J. Simmapim ◽  
D. Saennuad ◽  
T. Somaumchan
Keyword(s):  
Wave Equation ◽  
Solitary Wave ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
Rational Functions ◽  
Sixth Order ◽  
Mathematical Tool ◽  
Trigonometric Functions ◽  
Hyperbolic Functions ◽  
New Exact Solutions

The improved generalized tanh-coth method is used in nonlinear sixth-order solitary wave equation. This method is a powerful and advantageous mathematical tool for establishing abundant new traveling wave solutions of nonlinear partial differential equations. The new exact solutions consisted of trigonometric functions solutions, hyperbolic functions solutions, exponential functions solutions, and rational functions solutions. The numerical results were obtained with the aid of Maple.

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