Asymptotic Expansions of Fractional Derivatives and Their Applications
نویسندگان
چکیده
We compare the Riemann–Liouville fractional integral (fI) of a function f(z) with the Liouville fI of the same function and show that there are cases in which the asymptotic expansion of the former is obtained from those of the latter and the difference of the two fIs. When this happens, this fact occurs also for the fractional derivative (fD). This method is applied to the derivation of the asymptotic expansion of the confluent hypergeometric function, which is a solution of Kummer’s differential equation. In the present paper, the solutions of the equation in the forms of the Riemann–Liouville fI or fD and the Liouville fI or fD are obtained by using the method, which Nishimoto used in solving the hypergeometric differential equation in terms of the Liouville fD.
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