Fractional Calculus - Fractional Derivative of A Basic Power Function

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

\dfrac{d^{\frac{1}{2}}}{dx^{\frac{1}{2}}}x=\dfrac{\Gamma(1+1)}{\Gamma(1-\frac{1}{2}+1)}x^{1-\frac{1}{2}}=\dfrac{1!}{\Gamma(\frac{3}{2})}x^{\frac{1}{2}} = \dfrac{2x^{\frac{1}{2}}}{\sqrt{\pi}}.

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,

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