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Addition Formulas of leaf functions and Hyperbolic leaf functions

Computer Modeling in Engineering & Sciences 2020年第5期123卷 441-473页

作者： Kazunori Shinohara Department of Mechanical Systems Engineering Daido UniversityNagoya457-8530Japan

Addition formulas exist in trigonometric ***-angle and half-angle formulas can be derived from these ***,the relation equation between the trigonometric function and the hyperbolic function can be derived using an imaginary *** inverse hyperbolic function arsinher(r)■ro 1/√1+t^(2)dt p1tt2 dt is similar to the inverse trigonometric function arcsiner(r)■ro 1/√1+t^(2)dt p1t2 dt,such as the second degree of a polynomial and the constant term 1,except for the sign−and+.Such an analogy holds not only when the degree of the polynomial is 2,but also for higher *** such,a function exists with respect to the leaf function through the imaginary number i,such that the hyperbolic function exists with respect to the trigonometric function through this imaginary *** this study,we refer to this function as the hyperbolic leaf *** making such a definition,the relation equation between the leaf function and the hyperbolic leaf function makes it possible to easily derive various formulas,such as addition formulas of hyperbolic leaf functions based on the addition formulas of leaf *** the addition formulas,we can also derive the double-angle and half-angle *** then verify the consistency of these formulas by constructing graphs and numerical data.

关键词： leaf functions hyperbolic leaf functions lemniscate functions Jacobi elliptic functions ordinary differential equations nonlinear equations

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Lemniscate of leaf Function

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Computer Modeling in Engineering & Sciences 2021年第1期126卷 275-292页

作者： Kazunori Shinohara Department of Mechanical Systems Engineering Daido University10-3 Takiharu-choMinami-kuNagoya457-8530Japan

A lemniscate is a curve defined by two foci,F_(1) and F_(2).If the distance between the focal points of F_(1)−F_(2) is 2a(a:constant),then any point P on the lemniscate curve satisfy the equation PF_(1)·PF_(2)=a^(2).Jacob Bernoulli first described the lemniscate in *** Fagnano discovered the double angle formula of the lemniscate(1718).The Euler extended the Fagnano’s formula to a more general addition theorem(1751).The lemniscate function was subsequently proposed by Gauss around the year *** insights were summarized by Jacobi as the theory of elliptic functions.A leaf function is an extended lemniscate *** formulas of leaf functions have been presented in previous papers;these included the addition theorem of this function and its application to nonlinear *** this paper,the geometrical properties of leaf functions at n=2 and the geometric relation between the angle θ and lemniscate arc length l are presented using the lemniscate *** relationship between the leaf functions sleaf_(2)(l)and cleaf_(2)(l)is derived using the geometrical properties of the lemniscate,similarity of triangles,and the Pythagorean *** the literature,the relation equation for sleaf_(2)(l)and cleaf_(2)(l)(or the lemniscate functions,sl(l)and cl(l))has been derived analytically;however,it is not derived geometrically.

关键词： Geometry lemniscate of Bernoulli leaf functions lemniscate functions Pythagorean theorem triangle similarity

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