Fractional Derivative of A Basic Power Function
Let us assume that is a monomial of the form
The first derivative is as usual
Repeating this gives the more general result that
Which, after replacing the factorials with the Gamma function, leads us to
For and, we obtain the half-derivative of the function as
Repeating this process yields
which is indeed the expected result of
This extension of the above differential operator need not be constrained only to real powers. For example, the th derivative of the th derivative yields the 2nd derivative. Also notice that setting negative values for a yields integrals.
For a general function and, the complete fractional derivative is
For arbitrary, since the gamma function is undefined for arguments whose real part is a negative integer, it is necessary to apply the fractional derivative after the integer derivative has been performed. For example,
Read more about this topic: Fractional Calculus
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