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 ESSENTIAL TRIGONOMETRY WITHOUT GEOMETRY

2019 ◽  
Vol 71 (1) ◽  
Author(s):  
John Gresham ◽  
Bryant Wyatt ◽  
Jesse Crawford
Keyword(s):  
Unit Circle ◽  
Elementary Level ◽  
Comprehensive Treatment ◽  
Advanced Mathematics ◽  
Trigonometric Functions ◽  
Geometric Constructions ◽  
Rigorous Approach ◽  
Sine And Cosine Functions ◽  
Cosine Functions

Abstract The development of the trigonometric functions in introductory texts usually follows geometric constructions using right triangles or the unit circle. While these methods are satisfactory at the elementary level, advanced mathematics demands a more rigorous approach. Our purpose here is to revisit elementary trigonometry from an entirely analytic perspective. We will give a comprehensive treatment of the sine and cosine functions and will show how to derive the familiar theorems of trigonometry without reference to geometric definitions or constructions. Supplemental material is available for this article online.

scholarly journals Applying Computer Algebra Systems in Approximating the Trigonometric Functions

2018 ◽  
Vol 23 (3) ◽  
pp. 37 ◽  
Author(s):  
Le Quan ◽  
Thái Nhan
Keyword(s):  
Error Analysis ◽  
Computer Algebra ◽  
Numerical Algorithms ◽  
High Accuracy ◽  
Computer Algebra Systems ◽  
Trigonometric Functions ◽  
Modern Computer ◽  
Sine And Cosine Functions ◽  
Cosine Functions ◽  
Very High

We propose numerical algorithms which can be integrated with modern computer algebra systems in a way that is easily implemented to approximate the sine and cosine functions with an arbitrary accuracy. Our approach is based on Taylor’s expansion about a point having a form of kp, k∈Z and p=π/2, and being chosen such that it is closest to the argument. A full error analysis, which takes advantage of current computer algebra systems in approximating π with a very high accuracy, of our proposed methods is provided. A numerical integration application is performed to demonstrate the use of algorithms. Numerical and graphical results are implemented by MAPLE.

scholarly journals A Study on Some New Results Arising from (p,q)-Calculus

2018 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz ◽  
Serkan Araci
Keyword(s):  
Exponential Function ◽  
Chain Rule ◽  
Binomial Coefficients ◽  
Final Part ◽  
Trigonometric Functions ◽  
Elementary Functions ◽  
Exponential Functions ◽  
Hyperbolic Functions ◽  
Quantum Calculus ◽  
A Chain

This paper includes some new investigations and results for post quantum calculus, denoted by (p,q)-calculus. A chain rule for (p,q)-derivative is developed. Also, a new (p,q)-analogue of the exponential function is introduced and some its properties including the addition property for (p,q)-exponential functions are investigated. Several useful results involving (p,q)-binomial coefficients and (p,q)-antiderivative are discovered. At the final part of this paper, (p,q)-analogue of some elementary functions including trigonometric functions and hyperbolic functions are considered and some properties and relations among them are analyzed extensively.

scholarly journals SOME FORMULAS OF FRACTIONAL TRIGONOMETRIC FUNCTIONS

2021 ◽  
Author(s):  
CHII-HUEI CHII-HUEI
Keyword(s):  
Dirichlet Kernel ◽  
Trigonometric Functions ◽  
Sine And Cosine Functions ◽  
Cosine Functions

Abstract. This paper studies some properties of fractional trigonometric sine and cosine functions and we obtain the fractional Dirichlet kernel. The Mittag-Leffler function plays an important role in this article, and the results we obtained are the generalizations of formulas of the classical sine and cosine functions.

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 Bounds for the generalized Marcum function of the second kind

2021 ◽  
Author(s):  
Árpád Baricz ◽  
Nitin Bisht ◽  
Sanjeev Singh ◽  
V. Antony Vijesh
Keyword(s):  
Bessel Function ◽  
Bessel Functions ◽  
Modified Bessel Functions ◽  
Modified Bessel Function ◽  
Monotonicity Properties

AbstractIn this paper, we focus on the generalized Marcum function of the second kind of order $$\nu >0$$ ν > 0 , defined by $$\begin{aligned} R_{\nu }(a,b)=\frac{c_{a,\nu }}{a^{\nu -1}} \int _b ^ {\infty } t^{\nu } e^{-\frac{t^2+a^2}{2}}K_{\nu -1}(at)\mathrm{d}t, \end{aligned}$$ R ν ( a , b ) = c a , ν a ν - 1 ∫ b ∞ t ν e - t 2 + a 2 2 K ν - 1 ( a t ) d t , where $$a>0, b\ge 0,$$ a > 0 , b ≥ 0 , $$K_{\nu }$$ K ν stands for the modified Bessel function of the second kind, and $$c_{a,\nu }$$ c a , ν is a constant depending on a and $$\nu $$ ν such that $$R_{\nu }(a,0)=1.$$ R ν ( a , 0 ) = 1 . Our aim is to find some new tight bounds for the generalized Marcum function of the second kind and compare them with the existing bounds. In order to deduce these bounds, we include the monotonicity properties of various functions containing modified Bessel functions of the second kind as our main tools. Moreover, we demonstrate that our bounds in some sense are the best possible ones.

scholarly journals A Few Conditions for aC*-Algebra to Be Commutative

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Lajos Molnár
Keyword(s):  
Power Functions ◽  
Exponential Functions ◽  
Algebraic Properties ◽  
Sine And Cosine Functions ◽  
Cosine Functions

We present a few characterizations of the commutativity ofC*-algebras in terms of particular algebraic properties of power functions, the logarithmic and exponential functions, and the sine and cosine functions.

The Trigonometric Functions

1950 ◽  
Vol 43 (5) ◽  
pp. 187-192
Author(s):  
John W. Cell
Keyword(s):  
The Other ◽  
Advanced Mathematics ◽  
Trigonometric Functions ◽  
Points Of View ◽  
Sine And Cosine Functions ◽  
Cosine Functions

In this article we shall indicate many different methods by which the sine and cosine functions may be defined. (From these the other four functions may be obtained by their usual definitions in terms of the sine and cosine functions, viz., cot θ = cos θ/sin θ.) In the course of the discussion we shall consider the trigonometric functions from various points of view and we shall list properties which are not to be found in standard texts on trigonometry but which are found in advanced mathematics. We shall also indicate some general applications which are inherent in these various methods and sources for other applications.

Students' Conceptions of Sine and Cosine Functions When Representing Periodic Motion in a Visual Programming Environment

2018 ◽  
Vol 49 (4) ◽  
pp. 390-423 ◽  
Author(s):  
Anna F. DeJarnette
Keyword(s):  
Prior Knowledge ◽  
Secondary Students ◽  
Periodic Motion ◽  
Visual Programming ◽  
Programming Environment ◽  
Trigonometric Functions ◽  
Sine And Cosine Functions ◽  
Cosine Functions ◽  
Ferris Wheel

In support of efforts to foreground functions as central objects of study in algebra, this study provides evidence of how secondary students use trigonometric functions in contextual tasks. The author examined secondary students' work on a problem involving modeling the periodic motion of a Ferris wheel through the use of a visual programming environment. This study illustrates the range of prior knowledge and resources that students may draw on in their use of trigonometric functions as well as how the goals of students' work inform their reasoning about trigonometric functions.

Some Mellin-type Definite Integrals Involving Sine and Cosine Functions

2019 ◽  
Vol 10 (1) ◽  
pp. 222-237
Author(s):  
M. I. Qureshi ◽  
Kaleem A. Quraishi ◽  
Dilshad Ahamad
Keyword(s):  
Definite Integrals ◽  
Sine And Cosine Functions ◽  
Cosine Functions
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