In[]:=
(*Mathematica*)
In[]:=
(*Levydistributionfunction:builtin*)
In[]:=
f[x_,m_,s_]=
-
σ
2(x-μ)

3/2

σ
x-μ

2π
σ
/.μm/.σs
Out[]=
-
s
2(-m+x)

3/2

s
-m+x

2π
s
In[]:=
(*fractionalderivativeintegrationequation*)
In[]:=
(*https://www.wolframcloud.com/objects/demonstrations/AnOrdinaryFractionalDifferentialEquation-source.nb*)
In[]:=
dya[y_,t_,a_]=Integrate[(x-t)^(a-1)*y[t],{y,0,x}]
Out[]=
x
-0.877439-1.
(-t+x)
y[t],
xy[t]
0.877439
(-t+x)
,
xy[t]
0.877439
(-t+x)
,x
-0.877439+1.
(-t+x)
y[t]
In[]:=
(*SolvingtheLevydistributionasafractionalderivative*)
In[]:=
Solve
-
s
2(-m+x)

3/2

s
-m+x

2π
s
-x
-1+s
(-m+x)
y0,y
Out[]=
y-
-
s
2(-m+x)

3/2
-
s
m-x

(m-x)
-s
(-m+x)
2π
sx

In[]:=
(*testintegrationofLevyfractionalderivativegeneratorfunction*)
In[]:=
Integrate(x-m)^(s-1)*-
-
s
2(-m+x)

3/2
-
s
m-x

(m-x)
-s
(-m+x)
2π
sx
/.sN[Log[2]/Log[3]+I*Log[2]/Log[3]],{m,0,x}
Integrate
:Integral of -
0.398942
-
0.315465+0.315465
-1.m+x

0.63093+0.63093
-1.m+x
(m-1.x)x
does not converge on {0,x}.
Out[]=
x
∫
0
-
(0.316154-0.316154)
-
0.315465+0.315465
-m+x

3/2
-
0.63093+0.63093
m-x

(m-x)
x(-m+x)
m
In[]:=
(*vonKoch-Sierpinskicomplexdimension:complexsurfacefilling*)
In[]:=
FindRoot[Abs[Log[4]/Log[3]-x+I*Log[3]/Log[2]]-20,{x,0.1}]
Out[]=
{x0.042067}
In[]:=
(*Levygeneratorfunctionatcenterm=3:dimensions=N[Log[4]/Log[3]-0.042066954571888505+I*Log[3]/Log[2]]*)
In[]:=
N[Log[4]/Log[3]-0.042066954571888505+I*Log[3]/Log[2]]
Out[]=
1.21979+1.58496
In[]:=
ww=(x-m)^(s-1)*-
-
s
2(-m+x)

3/2
-
s
m-x

(m-x)
-s
(-m+x)
2π
sx
/.sN[Log[4]/Log[3]-0.042066954571888505+I*Log[3]/Log[2]]/.m3
Out[]=
-
(0.121657-0.158077)
-
0.609896+0.792481
-3+x

3/2
-
1.21979+1.58496
3-x
(3-x)
(-3+x)x
In[]:=
(*realplotofLevygeneratorfunction*)
In[]:=
(*blackbodyradiationprofile*)
In[]:=
g1=Plot[Re[ww],{x,3,10},PlotStyleRed,PlotRangeAll]
In[]:=
g1I=Plot[Im[ww],{x,3,10},PlotStyleRed,PlotRangeAll]
General
:Exp[-4265.01-5541.83] is too small to represent as a normalized machine number; precision may be lost.
In[]:=
NIntegrate[Re[ww],{x,3,Infinity}]
Out[]=
0.135779
In[]:=
NIntegrate[ww,{x,0,3}]
NIntegrate
:NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in x near {x} = {3.}. NIntegrate obtained 1.811842996006556×
414760405
10
+2.313570865341315×
414760405
10
 and 2.938602591561876×
414760405
10
for the integral and error estimates.
Out[]=
1.811842996006556×
414760405
10
+2.313570865341315×
414760405
10

In[]:=
g0=ComplexPlot3D[ww,{x,-3-4.5*I,6+4.5*I},ColorFunction"CyclicLogAbsArg",ImageSize1000,PlotPoints50,ViewPoint{5,5,5}]
In[]:=
Export["ComplexPlot3d_Levy_generator_function_complex_dimension_m3.jpg",{g1,g1I,g0}]
Out[]=
ComplexPlot3d_Levy_generator_function_complex_dimension_m3.jpg