On the Trigonometric and p-Trigonometric Functions of Elliptical Complex Variables

Year 2020, Volume: 3 Issue: 3, 143 - 154, 29.09.2020

Kahraman Esen Özen

https://doi.org/10.33434/cams.789085

Cited By: 3

Abstract

In the early 2000s, the geometry of a one-parameter family of generalized complex number systems was studied (Math. Mag. 77(2)(2004)). This family is denoted by Cp. It is well known that Cp matches up with the elliptical complex number system when p is any negative real number. By using this system, Özen and Tosun expressed the elliptical complex valued trigonometric functions cosine, sine and p-trigonometric functions p-cosine, p-sine (Adv. Appl. Clifford Algebras 28(3)(2018)). In this study, we introduce the remained elliptical complex valued trigonometric and p-trigonometric functions. Also we define the corresponding single-valued principal values of the inverse trigonometric and p-trigonometric functions by following the similar steps given in the literature.

Keywords

Generalized Complex Numbers, p-Trigonometric Functions, Elliptical Complex Numbers

References

• [1] I. M. Yaglom, Complex Numbers in Geometry, Academic Press, Newyork, 1968.
• [2] A. A. Harkin, J. B. Harkin, Geometry of generalized complex numbers, Math. Mag, 77(2) (2004), 118-129.
• [3] K. E. Özen, M. Tosun, p-Trigonometric approach to elliptic biquaternions, Adv. Appl. Clifford Algebr., 28(3) (2018), 62.
• [4] K. E. Özen, M. Tosun, Elliptic matrix representations of elliptic biquaternions and their applications, Int. Electron. J. Geom, 11 (2018), 96-103.
• [5] T. Erişir, M. A. Güngör, M. Tosun, The Holditch-type theorem for the polar moment of inertia of the orbit curve in the generalized complex plane, Adv. Appl. Clifford Algebr., 26 (2020), 1179-1193.
• [6] T. Erişir, M. A. Güngör, Holditch-type theorem for non-linear points in generalized complex plane Cp, Univers. J. Math. Appl., 1 (2018), 239-243.
• [7] Z. Derin, M. A. Güngör, On Lorentz transformations with elliptic biquaternions, Tbilisi Math., Sciendo (2020) 121-140.
• [8] F. S. Dündar, S. Ersoy, N. T. S. Pereira, Bobillier formula for the elliptical harmonic motion, An. St. Univ. Ovidius Constanta, 26 (2018), 103-110.
• [9] K. Eren, S. Ersoy, Burmester theory in Cayley-Klein planes with affine base, J. Geom, 109(3) (2018), 45.
• [10] K. Eren, S. Ersoy, Revisiting Burmester theory with complex forms of Bottema’s instantaneous invariants, Complex Var. Elliptic Equ., 62(4) (2017), 431-437.
• [11] I. A. Kosal, M. Öztürk, Best proximity points for elliptic generalized geraghty contraction mappings in elliptic valued metric spaces, AIP Conf. Proc, 2037(1) (2018), 020015.
• [12] N. B. Gürses, S. Yüce, One-parameter planar motions in generalized complex number plane CJ , Adv. Appl. Clifford Algebr., 25 (2015), 889-903.
• [13] K. E. Özen, On the elliptic biquaternions and their matrices, Sakarya University. Graduate School of Natural and Applied Sciences, Sakarya, Ph.D. Thesis; 2019.
• [14] M. Abramowitz, I. A. Stegun, Handbook of mathematical functions, Dover, New York, 1972.
• [15] A. I. Markushevich, Theory of functions of a complex variable, American Mathematical Soc., 2013.