fractional processes : derivative

derivatives were early defined integrally,
but we may interpolate the meaning to fractional

As with the counting numbers which by subtraction gave us the integers and by ratios became our rational numbers, mathematicians explore possible fractionation of countable processes: In particular here the derivative is defined as such a complete, unit, of process, and successive derivatives iterate the count: Negative derivatives indicate the inverse operation called, integration, of which the derivative is the original function; and fractional derivatives might indicate processes that can be multiply iterated to achieve the equivalent of one, full, unit of derivation (derivative).

For example, xδi c xn = c (n!/n-i!) xn-i . A fractional-derivative here, has fractional values for the derivative iterator, i ,- which works, upon inspection,- and in implementation, by the Sterling approximation to n! ...

xδi c xn ≈ c [(n+.5/e)(n+.5)/(n-i+.5/e)(n-i+.5)] xn-i =or= c [(n+.5)(n+.5)/(n-i+.5)(n-i+.5)] e-i xn-i .

Fractional derivatives have obvious utility, for example in hand-controlled motions, rovot finger-wiggle-pad control of computer-cursors, automobile cruise-drive controls,- where typically the center finger-range is absolute positional, the midway-out range is absolute velocity, and the edge is near or absolute acceleration: so that the user has continuous fine-to-fast-to-free control, without having to lift a finger back stroke the center for more. (Of course in practical design the temporal motion would be included in the operator equation, to avoid erratic detuned control on fingerprint grip-slip.)

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