Lorenzo Carl F, Hartley Tom T
National Aeronautics and Space Administration, Glenn Research Center, Cleveland, Ohio, USA.
Crit Rev Biomed Eng. 2008;36(1):57-78. doi: 10.1615/critrevbiomedeng.v36.i1.50.
The F-function, and its generalization the R-function, are of fundamental importance in the fractional calculus. It has been shown that the solution of the fundamental linear fractional differential equation may be expressed in terms of these functions. These functions serve as generalizations of the exponential function in the solution of fractional differential equations. Because of this central role in the fractional calculus, this paper explores various intrarelationships of the R-function, which will be useful in further analysis. Relationships of the R-function to the common exponential function, et, and its fractional derivatives are shown. From the relationships developed, some important approximations are observed. Further, the inverse relationships of the exponential function, et, in terms of the R-function are developed. Also, some approximations for the R-function are developed.
F 函数及其推广形式 R 函数在分数阶微积分中具有至关重要的地位。已经证明，基本线性分数阶微分方程的解可以用这些函数来表示。在分数阶微分方程的解中，这些函数是指数函数的推广形式。由于在分数阶微积分中的这一核心作用，本文探讨了 R 函数的各种内部关系，这将有助于进一步的分析。展示了 R 函数与普通指数函数 (e^t) 及其分数阶导数的关系。从所建立的关系中，可以观察到一些重要的近似。此外，还建立了指数函数 (e^t) 关于 R 函数的反关系。同时，也给出了一些 R 函数的近似。
