Transient-Time fractional-space trigonometry and application

Authors:
A. G. Radwan;Ahmed S. Elwakil
Affiliations:
Department of Engineering Mathematics, Cairo University, Egypt;Department of Electrical & Computer Engineering, University of Sharjah, Sharjah, UAE
Venue:
ICONIP'12 Proceedings of the 19th international conference on Neural Information Processing - Volume Part I
Year:
2012

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Abstract

In this work, we use the generalized exponential function in the fractional-order domain to define generalized cosine and sine functions. We then re-visit some important trigonometric identities and generalize 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 sin2(t)+cos2(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.