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