Transient-Time Fractional-Space Trigonometry and Application A.G. Radwan 1 and A.S. Elwakil 2,⋆ 1 Department of Engineering Mathematics, Cairo University, Egypt 2 Department of Electrical & Computer Engineering, University of Sharjah, P.O. Box 27272, Sharjah, Emirates {fagradwan,elwakil}@ieee.org Abstract. In this work, we use the generalized exponential function in the fractional-order domain to define generalized cosine and sine func- tions. We then re-visit some important trigonometric identities and gen- eralize them from the narrow integer-order subset to the more general fractional-order domain. It is clearly shown that trigonometric functions and trigonometric identities in the transient-time of a non-integer-order system have significantly different values from their steady-state values. Identities such as sin 2 (t) + cos 2 (t) = 1 are shown to be invalid in the transient-time of a fractional-order system. Some generalized hyperbolic functions and identities are also given in this work. Application to the evaluation of the step-response of a non-integer-order system is given. Keywords: Fractional-Calculus, Generalized Exponential function, Fractional order systems,Generalized Trigonometric Functions, Gener- alized Hyperbolic Functions. 1 Introduction Concepts of Fractional Calculus have been developed by mathematicians quite a long time ago [1, 2]. In particular, the Riemann-Liouville definition of the fractional derivative of a function f (t) is given by D α f (t)= D m ( J m-α f (t) ) , m - 1 <α ≤ m and the Laplace transform of the fractional derivative (left-inverse operator) is given by L (D α f (t)) = s α F (s) - m-1  k=0 D k J (m-α) f (0 + )s m-1-k , m - 1 <α ≤ m (1) Nevertheless, improvements and new relationships continue to be introduced in the literature [3]. It is unfortunate that these concepts have not found their way ⋆ Corresponding author. T. Huang et al. (Eds.): ICONIP 2012, Part I, LNCS 7663, pp. 40–47, 2012. c  Springer-Verlag Berlin Heidelberg 2012