In[]:=
(*Mathematica*)
In[]:=
(*Levydistributionfunction:builtin*)
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f[x_,m_,s_]=/.μm/.σs
-
σ
2(x-μ)
3/2
σ
x-μ
2π
σOut[]=
-
s
2(-m+x)
3/2
s
-m+x
2π
sIn[]:=
(*fractionalderivativeintegrationequation*)
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(*https://www.wolframcloud.com/objects/demonstrations/AnOrdinaryFractionalDifferentialEquation-source.nb*)
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dya[y_,t_,a_]=Integrate[(x-t)^(a-1)*y[t],{y,0,x}]
Out[]=
xy[t],,,xy[t]
-0.877439-1.
(-t+x)
xy[t]
0.877439
(-t+x)
xy[t]
0.877439
(-t+x)
-0.877439+1.
(-t+x)
In[]:=
(*SolvingtheLevydistributionasafractionalderivative*)
In[]:=
Solve-xy0,y
-
s
2(-m+x)
3/2
s
-m+x
2π
s-1+s
(-m+x)
Out[]=
y-(m-x)
-
s
2(-m+x)
3/2
-
s
m-x
-s
(-m+x)
2π
sxIn[]:=
(*testintegrationofLevyfractionalderivativegeneratorfunction*)
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Integrate(x-m)^(s-1)*-(m-x)/.sN[Log[2]/Log[3]+I*Log[2]/Log[3]],{m,0,x}
-
s
2(-m+x)
3/2
-
s
m-x
-s
(-m+x)
2π
sx0.398942
-
0.315465+0.315465
-1.m+x
0.63093+0.63093
-1.m+x
(m-1.x)x
Out[]=
x
∫
0
(0.316154-0.316154)(m-x)
-
0.315465+0.315465
-m+x
3/2
-
0.63093+0.63093
m-x
x(-m+x)
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(*vonKoch-Sierpinskicomplexdimension:complexsurfacefilling*)
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FindRoot[Abs[Log[4]/Log[3]-x+I*Log[3]/Log[2]]-20,{x,0.1}]
Out[]=
{x0.042067}
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(*Levygeneratorfunctionatcenterm=3:dimensions=N[Log[4]/Log[3]-0.042066954571888505+I*Log[3]/Log[2]]*)
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N[Log[4]/Log[3]-0.042066954571888505+I*Log[3]/Log[2]]
Out[]=
1.21979+1.58496
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ww=(x-m)^(s-1)*-(m-x)/.sN[Log[4]/Log[3]-0.042066954571888505+I*Log[3]/Log[2]]/.m3
-
s
2(-m+x)
3/2
-
s
m-x
-s
(-m+x)
2π
sxOut[]=
-
(0.121657-0.158077)(3-x)
-
0.609896+0.792481
-3+x
3/2
-
1.21979+1.58496
3-x
(-3+x)x
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(*realplotofLevygeneratorfunction*)
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(*blackbodyradiationprofile*)
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g1=Plot[Re[ww],{x,3,10},PlotStyleRed,PlotRangeAll]
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g1I=Plot[Im[ww],{x,3,10},PlotStyleRed,PlotRangeAll]
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NIntegrate[Re[ww],{x,3,Infinity}]
Out[]=
0.135779
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NIntegrate[ww,{x,0,3}]
414760405
10
414760405
10
414760405
10
Out[]=
1.811842996006556×+2.313570865341315×
414760405
10
414760405
10
In[]:=
g0=ComplexPlot3D[ww,{x,-3-4.5*I,6+4.5*I},ColorFunction"CyclicLogAbsArg",ImageSize1000,PlotPoints50,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