Abstract
Different forms of trigonometry have been proposed in the past to account for geometrical and applicative issues. Along with circular trigonometry, its hyperbolic counterpart has played a pivotal role to provide the geometrical framework of special relativity. The parabolic trigonometry is in between the previous two, and we discuss the relevant properties, point out the analogies with the standard forms and to the elementary problem of the projectile parabolic motion.
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Notes
It is also easy to infer the derivatives of the inverse TPF functions, namely \(\begin{array}{l} {\frac{d}{dx} {}_{p} c^{-1} (x)=-(1+x^{2} ),} \\ {\frac{d}{dx} {}_{p} s^{-1} (x)=\frac{2-x}{2\sqrt{1-x} } } \end{array}\)
References
Yaglom, I.M.: A Simple Non-euclidean Geometry and Its Physical Basis. Springer, Berlin (1979)
Davis, H.T.: Introduction to Non-linear Differential and Integral Equations. Dover Publications, New York (1962)
Harkin, A., Harkin, J.B.: Geometry of generalized complex numbers. Math. Mag. 77, 118–129 (2004)
Sobczyk, G.: New Foundations in Mathematics: The Geometric Concept of Number. Birkhauser, Basel (2013)
Ozdemir, M.: Introduction to hybrid numbers. Adv. Appl. Clifford Algebras (2018). https://doi.org/10.1007/s00006-018-0833-3
Dattoli, G., Licciardi, S., Sabia, E.: Generalized trigonometric functions and matrix parameterization. Int. J. Appl. Comput. Math. 3(Suppl 1), 115–128 (2017)
Ferrari, E.: Bollettino. UMI 18B, 933 (1981)
Edmunds, D.E., Gurka, P., Lang, J.: Properties of generalized trigonometric functions. J. Approx. Theor. 164, 47–56 (2012)
Gielis, J.: The Geometrical Beauty of Plants. Atlantis Press, Paris (2017)
Dattoli, G., Migliorati, M., Quattromini, M., Ricci, P.: The Parabolic Trigonometric Functions. arXiv:1102.1563v1 [math-ph]
Dattoli, G., Migliorati, M., Ricci, P.E.: The Parabolic Trigonometric Functions and Chebyshev Radicals, ENEA Report RT/2007/21/FIM (2007)
Nickalls, R.W.D.: A new approach to solving the cubic: Cardan’s solution revealed. Math. Gaz. 77, 354–359 (1993)
Witula, R., Slota, D.: Cardano’s formula, square roots, Chebyshev polynomials and radicals. J. Math. Anal. Appl. 363, 639–647 (2010)
Glasser, M.L.: Hypergeometric functions and the trinomial equation. J. Comput. Appl. Math. 118, 169 (2000)
Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. Dover Publications, New York (1964)
Goursat, E.A.: Course in Mathematical Analysis: Functions of a Complex Variable and Differential Equations, vol. 2, pp. 106–120. Dover Publications, New York (1959)
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Dattoli, G., Di Palma, E., Gielis, J. et al. Parabolic Trigonometry. Int. J. Appl. Comput. Math 6, 37 (2020). https://doi.org/10.1007/s40819-020-0789-6
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DOI: https://doi.org/10.1007/s40819-020-0789-6