Φ[N_][X_]:=-(N^2)+(N/2)Tr[Transpose[X].X]

ssmat[N_][v_]:=FoldList[{#1+#2,Take[v,{#1,#1+#2-1}]}&,1,Range[N]]

ssmat[3][{a,b,c}]

xx=
{
{0,a,b,c},
{0,0,d,e},
{0,0,0,f},
{0,0,0,0}
};

x4=xx-Transpose[xx]

Transpose[x4].x4//Tr

Skew

(x3={{0,a,b,d},{-a,0,c},{-b,-c,0}})//MatrixForm

Tr[Transpose[x3].x3]

Integrate[Exp[-β n (x^2)],{x,-∞,∞},Assumptions->{β>0,n>0}]

FullSimplify[((c^2)/((a^2)+(b^2)+(c^2)))-(((-(a^2)-(b^2)) (E^(-(Sqrt[-(a^2)-(b^2)-(c^2)]))))/(2 ((a^2)+(b^2)+(c^2))))-(((-(a^2)-(b^2)) (E^(Sqrt[-(a^2)-(b^2)-(c^2)])))/(2 ((a^2)+(b^2)+(c^2))))/.c->0/.b->0,Assumptions->{a>0}]

(a^2)+(b^2)

Clear[b]

Plot[-((b c)/((a^2)+(b^2)+(c^2)))-(((-b c+a (Sqrt[-(a^2)-(b^2)-(c^2)])) (E^(-(Sqrt[-(a^2)-(b^2)-(c^2)]))))/(2 ((a^2)+(b^2)+(c^2))))-(((-b c-a (Sqrt[-(a^2)-(b^2)-(c^2)])) (E^(Sqrt[-(a^2)-(b^2)-(c^2)])))/(2 ((a^2)+(b^2)+(c^2))))/.c->0/.b->0,{a,-20,20}]

{1,1,-1}.MatrixExp[x3][[1]]//FullSimplify

a=FullSimplify[(r^(n(n-1)/2-1)) Exp[-β n (r^2)]2(π^(n(n-1)/4))/Gamma[n(n-1)/4]/((((π^(1/2))/Sqrt[n β]))^(n(n-1)/2)),Assumptions->{β>0,n>0}]

b=Sum[a/.{r->π/2+π m},{m,0,5}];

LogLogPlot[b/.{β->10},{n,1,100},PlotRange->All]

FullSimplify[a/.r->π/2]

Integrate[a,{r,0,∞},Assumptions->{β>0,n>1}]

Series[a,{β,∞,1},Assumptions->{n>0,β>0}]

(((6s[1,1,1]-s[0,1,1]-s[1,0,1]-s[1,1,0]-s[2,1,1]-s[1,2,1]-s[1,1,2]))^2)//Expand

7

((s[0,1,1])^2)+cl+((s[1,0,1])^2)+2 s[0,1,1] s[1,1,0]+2 s[1,0,1] s[1,1,0]+((s[1,1,0])^2)-12 s[0,1,1] s[1,1,1]-12 s[1,0,1] s[1,1,1]-12 s[1,1,0] s[1,1,1]+36 ((s[1,1,1])^2)+2 s[0,1,1] s[1,1,2]+2 s[1,0,1] s[1,1,2]+2 s[1,1,0] s[1,1,2]-12 s[1,1,1] s[1,1,2]+((s[1,1,2])^2)+2 s[0,1,1] s[1,2,1]+2 s[1,0,1] s[1,2,1]+2 s[1,1,0] s[1,2,1]-12 s[1,1,1] s[1,2,1]+2 s[1,1,2] s[1,2,1]+((s[1,2,1])^2)+2 s[0,1,1] s[2,1,1]+2 s[1,0,1] s[2,1,1]+2 s[1,1,0] s[2,1,1]-12 s[1,1,1] s[2,1,1]+2 s[1,1,2] s[2,1,1]+2 s[1,2,1] s[2,1,1]+((s[2,1,1])^2)

A=Sqrt[1-((8g)/(b^2))η[x]δFoδη];

Series[Sqrt[1-((8g)/(b^2))η[x]δFoδη],{η[x],0,1}]

Series[((b^2)/(4g))(A-(A^2)),{η[x],0,1}]

cc=CC-b(2-A)((1/(b A χ))

Clear[A]

A[g

CCC[CC_,b_,g_,

FourierTransform[1-((4 (g δFoδη) η[x])/(b^2)),

A=Sqrt[1-((8g)/(b^2))η[x](2 g ((ϵ[x])^2)η[x]-b ϵ[x])]/.ss[[1]]/.η[x]->ηs Cos[qs x]//Simplify

AA=(Sqrt[((1+12 b ((Cos[qs x])^2)+4 (b^2)((Cos[qs x])^4))/(((1+2 b ((Cos[qs x])^2)))^2))])

FourierTransform[AA,x,k,Assumptions->{b>0,k>0,qs>0}]

Convolve[(1/(d((((qs^2)-(q^2)))^2)+Δr)),Sqrt[1-((8g)/(b^2))

ss=Solve[0==CC ϵ[x]-b η[x]+2 g ϵ[x] ((η[x])^2),ϵ[x]]