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,

Read more about this topic:  Fractional Calculus

Famous quotes containing the words fractional, derivative, basic, power and/or function:

    “Hummingbird
    stay for a fractional sharp
    sweetness, and’s gone, can’t take
    more than that.”
    —Denise Levertov (b. 1923)

    “Poor John Field!—I trust he does not read this, unless he will improve by it,—thinking to live by some derivative old-country mode in this primitive new country.... With his horizon all his own, yet he a poor man, born to be poor, with his inherited Irish poverty or poor life, his Adam’s grandmother and boggy ways, not to rise in this world, he nor his posterity, till their wading webbed bog-trotting feet get talaria to their heels.”
    —Henry David Thoreau (1817–1862)

    “Southerners, whose ancestors a hundred years ago knew the horrors of a homeland devastated by war, are particularly determined that war shall never come to us again. All Americans understand the basic lessons of history: that we need to be resolute and able to protect ourselves, to prevent threats and domination by others.”
    —Jimmy Carter (James Earl Carter, Jr.)

    “Even voting for the right is doing nothing for it. It is only expressing feebly your desire that it should prevail. A wise man will not leave the right to the mercy of chance, nor wish it to prevail through the power of the majority.”
    —Henry David Thoreau (1817–1862)

    “Nobody seriously questions the principle that it is the function of mass culture to maintain public morale, and certainly nobody in the mass audience objects to having his morale maintained.”
    —Robert Warshow (1917–1955)