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Re: Trying to define: Fractional Derivatives & Leibniz' display form for output and templates
*To*: mathgroup at smc.vnet.net
*Subject*: [mg22881] Re: Trying to define: Fractional Derivatives & Leibniz' display form for output and templates
*From*: John Doty <jpd at w-d.org>
*Date*: Mon, 3 Apr 2000 00:04:11 -0400 (EDT)
*References*: <8bhvta$noq@smc.vnet.net> <8c879v$736@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
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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