New approach to the fractional derivatives

We introduce a new approach to the fractional derivatives of the analytical functions using the Taylor series of the functions. In order to calculate the fractional derivatives of f , it is not sufficient to know the Taylor expansion of f , but we should also know the constants of all consecutive integrations of f . For example, any fractional derivative of e x is e x only if we assume that the n th consecutive integral of e x is e x for each positive integer n . The method of calculating the fractional derivatives very often requires a summation of divergent series, and thus, in this note, we first introduce a method of such summation of series via analytical continuation of functions.

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