fractional processes : derivative (continuum calculus)

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 and by integral roots and powers (i.e. rational-powers) became extensions of rationals and beyond were the infinitesimals and irrationals and reals combining them all, mathematicians explore fractionation of countable processes... In particular here, the derivative, was defined as such, a complete unit of process, and successive derivatives iterated the count, (and likewise the polynomials upon which the derivatives operate, were, of integral powers), and negative derivatives indicated the inverse, operation, called, antiderivative, or integral (integration), of which the derivative is the original function... So now, fractional derivatives will indicate processes that can be multiply iterated (compounded) inverse-fractionally to achieve the equivalent of one, full unit of derivative (derivation), e.g. half-derivatives by composition δ˝ δ˝ = δ1 the full-derivative (in functional/operator-notation; derivative is an exponential-like process; likewise factorial, is exponential-like)—

Building by inspection: the k-derivative δk xn = (n!/n-k!) xn-k δ x (which gives a little insight to the basic process of functions relating inputs and outputs, not just doing something mathematical on inputs), works upon inspection by integer values, both derivative and anti- (except the infamous δ ln x = 1/x 'because' δ x0 = 0/x as x0 = 1 *); and now in full fractional-implementation for all Real values of the derivative iterator k, with fractional-derivatives having fractional-value-processes, (and, in approximation, by our faster-than-Stirling of n!)…

* (this was always a musing point, that δ xΔ/Δ = xΔ-1 → 1/x = δ ln x , but meanwhile xΔ/Δ → 1/Δ → ∞ as near-'constant')

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 central range is pure positional, the ⅓-out range is pure velocity, the ⅔-out range is pure acceleration, and the edge is pure pump, so that the user has continuous fine-to-fast-to-free control,without having to lift a finger back stroke to anywhere 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, and equipment vibrations, etc.)

[2018] Sidebar/footnote: the notion of fractionation of integral processes extends into the realm of microcomputation where arithmetic instruction operations are finely split into substages, individually buffered and sequenced to equal one whole 'pipelined' stage, while thus effecting many, replicates of one ‘expensive’ hardware ALU, by multiple ‘inexpensive’ buffers, (the tradeoff of speed, power, hardware)…

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