In this paper, we derive an analytical exact form of signum function, which evidently constitutes a fundamental concept of Communication and Control Theory along with Digital Control Systems and is also involved in many other fields of applied mathematics and engineering.

In particular, this important function is exhibited in a simple manner as a finite combination of five inverse tangent functions. The novelty of this work when compared to other analytical expressions of this function is that the proposed explicit representation is not performed in terms of miscellaneous special functions, such as Bessel functions, Error function, and Beta function, and also is neither the limit of a function, nor the limit of a sequence of functions with a point-wise or uniform convergence.

Hence, this formula may be much more appropriate and useful in the computational procedures which are inserted into Control Theory techniques and other engineering practices.

signum function, singularity function, inverse trigonometric functions, finite summation.

Received: March 21, 2024;  Accepted: April 24, 2024; Published: May 11, 2024

How to cite this article: J. Venetis, Αn exact form of signum function, Advances and Applications in Discrete Mathematics 41(4) (2024), 311-324. https://doi.org/10.17654/0974165824022

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:

[1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover Publications INC, 1972.
[2] L. Berg, Introduction to the Operational Calculus, North-Holland Publishing Company, 1967.
[3] R. P. Kanwal, Generalized Functions Theory and Technique: Theory and Technique, 2nd ed., Boston, MA: Birkhäuser, 1998.
[4] R. Priemer, Introductory Signal Processing (Advanced Electrical and Computer Engineering) (Vol. 6), World Scientific Publishing Company, 1990.
[5] P. Senthilkumaran, Singularities in Physics and Engineering, IOP Publishing Ltd., 2024.
[6] J. Mathews and R. Howell, Complex Analysis for Mathematics and Engineering, Jones & Bartlett Publishers, 2012.
[7] K. Knopp, Singularities, Section IV in Theory of Functions, Parts I and II, Two Volumes Bound as One, Part I, New York, Dover, 1996, p. 117-139.
[8] S. G. Krantz, Removable singularities, poles, and essential singularities, Chapter in Handbook of Complex Variables, Boston, MA: Birkhäuser, 1999, p. 42.
[9] F. O. Simons, R. C. Harden and A. L. Robinson, Continuous singularity functions insight, extensions, and corrections, Proceedings of 28th Southeastern Symposium on System Theory, IEEE, 1996, pp. 447-451.
[10] H. Oukhaba, Sign functions of imaginary quadratic fields and applications, Annales de l’institut Fourier 55(3) (2005), 753-772.
[11] Dan Ciulin, About sign function and some extensions, Innovations and Advanced Techniques in Systems, Computing Sciences and Software Engineering, Dordrecht: Springer Netherlands, 2008, pp. 148-153.
[12] Beong In Yun and Miodrag S. Petković, Iterative methods based on the signum function approach for solving nonlinear equations, Numerical Algorithms 52(4) (2009), 649-662.
[13] Jian Wu, Leang S. Shieh and Jason S. H. Tsai, A scalar sign function approach to digital control of continuous-time chaotic systems with state constraints, International Journal of Bifurcation and Chaos 19(06) (2009), 2009-2029.
[14] Jian Zhang, Yongpeng Zhang and Warsame Ali, Linearization modeling for non-smooth dynamical systems with approximated scalar sign function, 2011 50th IEEE Conference on Decision and Control and European Control Conference, IEEE, 2011.
[15] Vladimir V. Podolskii, Degree-uniform lower bound on the weights of polynomials with given sign function, Proceedings of the Steklov Institute of Mathematics 274(1) (2011), 231-246.
[16] A. Frommer, K. Kahl, T. Lippert and H. Rittich, Error bounds for the sign function, 2011. arXiv preprint arXiv:1112.3482.
[17] Peter Cariani, Sign functions in natural and artificial systems, International Handbook of Semiotics (2015), 917-949.
[18] P. Benner and J. Deniben, Numerical solution to low rank perturbed Lyapunov equations by the sign function method, PAMM 16(1) (2016), 723-724.
[19] S. Zhang, X. Wang and Z. Zeng, A simple no-equilibrium chaotic system with only one signum function for generating multidirectional variable hidden attractors and its hardware implementation, Chaos: An Interdisciplinary Journal of Nonlinear Science 30(5) (2020), 129-142.
[20] J. Venetis, An analytic exact form of the unit step function, Mathematics and Statistics 2(7) (2014), 235-237.
[21] J. Venetis, An analytic exact form of the Heaviside step function, Advances and Applications in Discrete Mathematics 22(2) (2019), 153-159.
[22] M. Kyle Murphy, Explicit Forms of Discontinuous Functions the Dirac Delta and Irreducible Forms (preprint).
https://www.academia.edu/11704122/Explicit_Forms_of_Discontinuous_ Functions_the_Dirac_Delta_and_Irreducible_Forms.
[23] J. C. Venetis, An explicit expression of the unit step function, International Review of Electrical Engineering 18(1) (2023), 83-87.
[24] J. C. Venetis, An explicit form of ramp function, Applied Math 4(2) (2024), 442-451.
[25] Clement Vavasor Durell and Alan Robson, Advanced Trigonometry, Courier Corporation, 2003.
[26] Gilbert Strang, Calculus, Vol. 1, SIAM, 1991.
[27] I. A. Maron, Problems in Calculus of One Variable, with Elements of Theory, Translated from the Russian by Leonid Levant [Revised from the 1970 Russian Ed.] Mir Publishers, 1973.
[28] Stanley I. Grossman, Calculus, Academic Press, 2014.