This paper reviews some properties of the gamma function, particularly the incomplete gamma function and its complement, as a function of the Laplace variable . The utility of these functions in the solution of initialization problems in fractional-order system theory is demonstrated. Several specific differential equations are presented, and their initialization responses are found for a variety of initializations. Both the time-domain and Laplace-domain solutions are obtained and compared. The complementary incomplete gamma function is shown to be essential in finding the Laplace-domain solution of a fractional-order differential equation.
Application of Incomplete Gamma Functions to the Initialization of Fractional-Order Systems
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Hartley, T. T., and Lorenzo, C. F. (March 11, 2008). "Application of Incomplete Gamma Functions to the Initialization of Fractional-Order Systems." ASME. J. Comput. Nonlinear Dynam. January 2008; 3(2): 021103. https://doi.org/10.1115/1.2833480
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