Refinements of Some Classical Inequalities Involving Sinc and Hyperbolic Sinc Functions
1
Citation
20
Reference
10
Related Paper
Citation Trend
Abstract:
Abstract Several bounds of trigonometric-exponential and hyperbolic-exponential type for sinc and hyperbolic sinc functions are presented. In an attempt to generalize the results, some known inequalities are sharpened and extended. Hyperbolic versions are also established, along with extensions.Keywords:
Sinc function
Many engineering problems can be solved by methods involving complex numbers and complex functions. In the definitions below we will prove the relationship between trigonometric functions and hyperbolic functions, where the hyperbolic function is an extension of the trigonometric function.
Keywords: trigonometric functions; hyperbolic functions
Trigonometric integral
Trigonometric substitution
Pythagorean trigonometric identity
Cite
Citations (0)
Sine
Inverse trigonometric functions
Cite
Citations (35)
The restrictions imposed on the solutions of nonuniform transmission lines, the impedance of which can be described by various trigonometric and hyperbolic functions, have been relaxed here to encompass additional possibilities. These possibilities also allow the nonuniform impedance to be represented by the corresponding reciprocals of the trigonometric and hyperbolic functions considered by Holt and Bowron.
Trigonometric integral
Trigonometric substitution
Cite
Citations (1)
The energy levels of the time-independent Schrödinger equation are computed in three dimensions by applying double exponential Sinc collocation method. Numerical results are provided to demonstrate the high accuracy of the proposed approach for different potential functions. Comparative tests with the single exponential Sinc collocation method are made to confirm the superiority of the double exponential Sinc collocation method.
Sinc function
Collocation (remote sensing)
Energy method
Cite
Citations (0)
Abstract Several bounds of trigonometric-exponential and hyperbolic-exponential type for sinc and hyperbolic sinc functions are presented. In an attempt to generalize the results, some known inequalities are sharpened and extended. Hyperbolic versions are also established, along with extensions.
Sinc function
Cite
Citations (1)
According to two dependent rational solutions to a generalized Riccati equation together with the equation itself, a rational-exponent solution to a nonlinear partial differential equation can be constructed. By selecting different parameter values in the rational-exponent solution, many families of combinatorial solutions combined with a rational function such as hyperbolic functions or trigonometric functions, are rapidly derived. This method is applied to the Whitham?Broer?Kaup equation and a series of combinatorial solutions are obtained, showing that this method is a more concise and efficient approach and can uniformly construct many types of combined solutions to nonlinear partial differential equations.
Cite
Citations (1)
Sinc function
Series expansion
Error function
Cite
Citations (22)
Periodic wave
Cite
Citations (16)
In this article, the exact solutions of the stochastic conformable Broer–Kaup equations with conformable derivatives which describe the bidirectional propagation of long waves in shallow water are obtained using the modified exponential function method and the generalized Kudryashov method. These exact solutions consist of hyperbolic, trigonometric, rational trigonometric, rational hyperbolic, and rational function solutions, respectively. This shows that the proposed methods are competent and sufficient. In addition, it is aimed to better understand the physical properties by drawing two- and three-dimensional graphics of the exact solutions according to different parameter values. When these exact solutions obtained by two different methods are compared with the solutions attained by other methods, it can be said that these two methods are competent.
Conformable matrix
Cite
Citations (1)
This chapter contains sections titled: Introduction Relation Between Exponential and Trigonometric Functions Similarities and Differences in the Behavior of Hyperbolic and Circular Functions Derivatives of Hyperbolic Functions Curves of Hyperbolic Functions The Indefinite Integral Formulas for Hyperbolic Functions Inverse Hyperbolic Functions Justification for Calling sinh and cosh as Hyperbolic Functions Just as sine and cosine are Called Trigonometric Circular Functions
Hyperbolic angle
Sine
Trigonometric integral
Inverse trigonometric functions
Trigonometric substitution
Ultraparallel theorem
Cite
Citations (0)