Addition Formulas of Leaf Functions and Hyperbolic Leaf Functions

Shinohara, Kazunori

doi:10.32604/cmes.2020.08656

Cited by 1 publication

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“…Leaf functions are extended lemniscate functions. Various formulas for leaf functions such as the addition theorem of the leaf functions and its application to nonlinear equations have been presented [30][31][32].…”

Section: Originality and Purposementioning

confidence: 99%

Lemniscate of Leaf Function

Shinohara¹

2021

Computer Modeling in Engineering &Amp; Sciences

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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 1694. The 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 1800. These insights were summarized by Jacobi as the theory of elliptic functions. A leaf function is an extended lemniscate function. Some formulas of leaf functions have been presented in previous papers; these included the addition theorem of this function and its application to nonlinear equations. In 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 curve. The 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 theorem. In 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.

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Section: Originality and Purposementioning

confidence: 99%