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3. Powers

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The case of powers of x also has some simplicity that allows its generalization. The case of integer order derivatives

 (3.1)

can be easily generalized to non-integer order derivatives

 (3.2)

which can be applied to any function that can be expanded in powers of x

 (3.3)

Expanding the function in Taylor series,

 (3.4)

or expanding it in Laurent series,

 (3.5)

This can be an useful tool for calculating fractional derivatives. However, we should compare these results of powers with the previous results of exponentials to see if they agree. With the result of exponentials (2.1),

 (3.6)

but with the result of powers (3.4),

 (3.7)

If we compare both results, we see that they only agree for integer values of a. We shall see later where these discrepancies come from, and how they can be avoided.