On Tuesday, 14 January 2014 00:54:13 UTC+11, Roger Bagula wrote:

1) Fractional derivatives are connected to fractal dimension
2) Mandelbrot's negative fractal dimensions exist
3) Lapidus' complex dimensions exist
Therefore logic demands that
Complex fractional derivatives exist and are connected
fractional dimensions.
A little thinking on how complex dimensions are connected
to affine transforms and we have that self-similar fractals like the Cantor set can be converted to affine fractal sets by a
complex fractional derivative.
Mathematica:( took me 5 tries!)
In[221]:= Clear[f, dlst, pt, cr, ptlst, M, r, p, rotate, g]

In[222]:= (* by Roger Bagula © sunday January 12, 2014*)

In[223]:= g[x_, m_, q_] = Gamma[m + 1]/Gamma[m - q + 1] x^(m - q)

Out[223]= (x^(m - q) Gamma[1 + m])/Gamma[1 + m - q]

In[224]:= N[g[x/3, 1, N[1/4 + I*Sqrt[3]/4]]]

Out[224]= (0.425523 + 0.285079 I) x^(0.75 - 0.433013 I)

In[225]:= s /.
NSolve[4*((0.425523370922056` + 0.28507927066159416` I)) ^s == 1,
s][[1]]

During evaluation of In[225]:= NSolve::ifun: Inverse functions are being used by NSolve, so some solutions may not be found; use Reduce for complete solution information. >>

Out[225]= 1.16513 + 1.02793 I

In[226]:= Abs[%]

Out[226]= 1.55376

In[227]:= Clear[f]

In[228]:= dlst = Table[ Random[Integer, {1, 4}], {n, 500000}];
f[1, {x_, y_}] =
N[{Re[(0.425523370922056` + 0.28507927066159416` I) x^(
0.75` - 0.4330127018922193` I) ] -
Im[(0.425523370922056` + 0.28507927066159416` I) y^(
0.75` - 0.4330127018922193` I) + 1/2],
Im[(0.425523370922056` + 0.28507927066159416` I) x^(
0.75` - 0.4330127018922193` I) ] +
Re[(0.425523370922056` + 0.28507927066159416` I) y^(
0.75` - 0.4330127018922193` I) + 1/2]}];
f[2, {x_, y_}] =
N[{Re[(0.425523370922056` + 0.28507927066159416` I) x^(
0.75` - 0.4330127018922193` I) + 1/2] -
Im[(0.425523370922056` + 0.28507927066159416` I) y^(
0.75` - 0.4330127018922193` I) ],
Im[(0.425523370922056` + 0.28507927066159416` I) x^(
0.75` - 0.4330127018922193` I) + 1/2] +
Re[(0.425523370922056` + 0.28507927066159416` I) y^(
0.75` - 0.4330127018922193` I) ]}];
f[3, {x_, y_}] =
N[{Re[(0.425523370922056` + 0.28507927066159416` I) x^(
0.75` - 0.4330127018922193` I)] -
Im[(0.425523370922056` + 0.28507927066159416` I) y^(
0.75` - 0.4330127018922193` I)],
Im[(0.425523370922056` + 0.28507927066159416` I) x^(
0.75` - 0.4330127018922193` I)] +
Re[(0.425523370922056` + 0.28507927066159416` I) y^(
0.75` - 0.4330127018922193` I) ]}];
f[4, {x_, y_}] =
N[{Re[(0.425523370922056` + 0.28507927066159416` I) x^(
0.75` - 0.4330127018922193` I) + 1/2] -
Im[(0.425523370922056` + 0.28507927066159416` I) y^(
0.75` - 0.4330127018922193` I) + 1/2],
Im[(0.425523370922056` + 0.28507927066159416` I) x^(
0.75` - 0.4330127018922193` I) + 1/2] +
Re[(0.425523370922056` + 0.28507927066159416` I) y^(
0.75` - 0.4330127018922193` I) + 1/2]}];

pt = {0.5, 0.5};
cr[n_] :=
Flatten[Table[
If[i == j == k == 1, {}, RGBColor[i, j, k]], {i, 0, 1}, {j, 0,
1}, {k, 0, 1}]][[1 + Mod[n, 7]]];
ptlst[n_] :=
Table[{cr[dlst[[j]]], Point[pt = f[dlst[[j]], Sequence[pt]]]},
{j, Length[dlst]}];

In[236]:= Show[Graphics[Join[{PointSize[.001]}, ptlst[n]]],
AspectRatio -> Automatic, ImageSize -> 1000]

yes, connect, but how Fractional derivatives are connected to fractal dimension? what relationship?

reference math handbook
www.mathHandbook.com

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)