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In[]:=
deploy
Thu 26 Aug 2021 11:50:03
Series expansion used in https://math.stackexchange.com/questions/4228308/average-value-of-fracxa2xxa3x-over-surface-of-n-dimensional-sphere
Utility from https://mathematica.stackexchange.com/a/15025/217
In[]:=
multiTaylor[f_,{vars_?VectorQ,pt_?VectorQ,n_Integer?NonNegative}]:=Sum[Nest[(vars-pt).#&,(D[f,{vars,k.}]/.Thread[vars->pt]),k.]/k.!,{k.,0,n},Method->"Procedural"]
In[]:=
expand[order_]:=Clear[A,B,muA,muB];n=2;xs=Array[x,n];Ex[expr_]:=NExpectation[expr,(x[#]NormalDistribution[]&/@Range[n])];series=multiTaylor,{{A,B},{muA,muB},order};A=Total@Table,{i,1,n};B=Total@Table,{i,1,n};muA=Ex[A];muB=Ex[B];Ex[series];orders=Table[k,{k,0,4}];vals=Table[expand[k],{k,orders}];ListPlot[vals,FillingAxis,PlotRangeAll]TableForm[vals,TableHeadings{orders,{"Taylor expansion","simplified"}}]
A
B
2
x[i]
2
i
2
x[i]
3
i
Out[]=
Out[]//TableForm=
0 | 1.11111 |
1 | 1.1123 |
2 | 1.26475 |
3 | 0.650206 |
4 | 4.713 |