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Re: Trying to define: Fractional Derivatives & Leibniz' display form for output and templates


The problem cries out for "Generalized Functions", but unfortunately
Mathematica's support for these is shallow and quirky. Try:

genD[func_, {var_, order_}] :=
  InverseFourierTransform[
   (-(I*freq))^order*
    FourierTransform[func, var,
     freq], freq, var]

This should behave like D[], but allow fractional and negative order. It works
in some simple cases (but you may need to coax it to give you the form you
want):

Simplify[ExpToTrig[genD[Sin[x],
    {x, 1/2}]]]

Yields:

Cos[x] + Sin[x]
---------------
    Sqrt[2]

but:

genD[x^2, {x, 1/2}]

yields:

"Indeterminate expression ComplexInfinity+ComplexInfinity encountered."

and:

genD[x^n, {x, 1/2}]

yields a fascinating expression that looks at least partly sensible, but is
zero for integer values of n.

On a related note, the following is rather entertaining:

Table[FourierTransform[x^n, x, f], {n, -2, 2, 1/2}] -
  Table[Evaluate[FourierTransform[x^n, x, f]], {n, -2, 2, 1/2}]

The order of the evaluation "shouldn't make a difference", but in fact only one
element of the resulting table is 0 (and even in that case, you need Simplify[]
to get there).

--
John Doty  "You can't confuse me, that's my job."
Home: jpd at w-d.org
Work: jpd at space.mit.edu




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