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)

    “Justice begins with the recognition of the necessity of sharing. The oldest law is that which regulates it, and this is still the most important law today and, as such, has remained the basic concern of all movements which have at heart the community of human activities and of human existence in general.”
    —Elias Canetti (b. 1905)

    “It recognizes no morality but a sham morality meant for deceit, no honor even among thieves and of a thievish sort, no force but physical force, no intellectual power but cunning, no disgrace but failure, no crime but stupidity.”
    —Woodrow Wilson (1856–1924)

    “The more books we read, the clearer it becomes that the true function of a writer is to produce a masterpiece and that no other task is of any consequence.”
    —Cyril Connolly (1903–1974)