Design and analysis of the sine and cosine functions in fractional order case

This paper systematically investigates the trigonometric functions from the exponential function point of view. After recalling the classical exponential function, the Mittag-Leffler function is introduced as a generalization. Then, the cosine function and sine function are extended to the fractional order case. With the introduction of nabla discrete time case, a corresponding exponential function is defined. By using the $N$-transform, the discrete time cosine function and sine function are designed and analyzed both in the integer order and the fractional order cases. To solve the periodic oscillation problem, the poles are specially chosen and laid in the marginal position. Illustrative examples are provided to validate the elaborated results.