Mat=(({{M, 0, 0, 0, 0, 0, 0, 0, 0, 2I v1}, {0, M, 0, 0, 0, 0, 0, 0, 0, 2 I v2}, {0, 0, M, 0, 0, 0, 0, 0, 0, 2I v3}, {0, 0, 0, M, 0, 0, 0, 0, 0, 2I v4}, {0, 0, 0, 0, M, 0, 0, 0, 0, 2I v5}, {0, 0, 0, 0, 0, M, 0, 0, 0, 2I v6}, {0, 0, 0, 0, 0, 0, M, 0, 0, 2I v7}, {0, 0, 0, 0, 0, 0, 0, M, 0, 2I v8}, {0, 0, 0, 0, 0, 0, 0, 0, M, 2I v9}, {-2I v1, -2I v2, -2I v3, -2I v4, -2I v5, -2I v6, -2I v7, -2 I v8, -2I v9, M}}));

Eigenvalues[Mat]

Mat=(({{M, 0, 0, 0, 0, 2I v1}, {0, M, 0, 0, 0, 2 I v2}, {0, 0, M, 0, 0, 2I v3}, {0, 0, 0, M, 0, 2I v4}, {0, 0, 0, 0, M, 2I v5}, {-2I v1, -2I v2, -2I v3, -2I v4, -2I v5, M}}));

Eigenvalues[Mat]

H=((((((2π m)/β))^2+r^2+v)^1((((2π m)/β))^2+r^2-v)^1)/(((((2π m)/β))^2+r^2)^0((((2π m)/β))^2+r^2+2v)((((2π m)/β))^2+r^2-2v)));

Hexp=Simplify[Series[H,{v,0,4}]]

H=((((((2π m)/β))^2+r^2+v)^2((((2π m)/β))^2+r^2-v)^2)/(((((2π m)/β))^2+r^2)^2((((2π m)/β))^2+r^2+2v)((((2π m)/β))^2+r^2-2v)));

Hexp=Simplify[Series[H,{v,0,4}]]

Mat=(({{M, 0, 0, 0, 0, 0, 0, 0, 0, 2 v1}, {0, M, 0, 0, 0, 0, 0, 0, 0, 2 v2}, {0, 0, M, 0, 0, 0, 0, 0, 0, 2v3}, {0, 0, 0, M, 0, 0, 0, 0, 0, 2v4}, {0, 0, 0, 0, M, 0, 0, 0, 0, 2v5}, {0, 0, 0, 0, 0, M, 0, 0, 0, 2v6}, {0, 0, 0, 0, 0, 0, M, 0, 0, 2v7}, {0, 0, 0, 0, 0, 0, 0, M, 0, 2v8}, {0, 0, 0, 0, 0, 0, 0, 0, M, 0}, {-2v1, -2v2, -2v3, -2v4, -2v5, -2v6, -2v7, -2v8, 0, M}}));

Eigenvalues[Mat]

H=((((((2π m)/β))^2+r^2+v)^4((((2π m)/β))^2+r^2-v)^4)/(((((2π m)/β))^2+r^2)^6((((2π m)/β))^2+r^2+2v)((((2π m)/β))^2+r^2-2v)));

Hexp=Simplify[Series[H,{v,0,4}]]

H4=SeriesCoefficient[Hexp,4]

sumH4=Simplify[Sum[H4,{m,-∞,+∞}]]

Limit[r^7sumH4/.{r->(x/β)},x->∞]

H4=SeriesCoefficient[Hexp,4]/.{m^2->m^2+n^2}

tmp=Sum[(1/((1+((m^2+n^2)/(r^2)))^4)),{m,-∞,+∞}]

Limit[(1/r)tmp,r->∞]

Simplify[Sin[2x]Tan[x]]

Expand[TrigToExp[16Sin[(x/2)]^4]]

Integrate[σ^((5-p)/2)Exp[-I σ r^2],{σ,0,+∞}]

144.55+17.34+19.99

tmp=Sum[(1/((1+4((m^2)/(r^2))))),{m,-∞,+∞}]

Limit[(1/r)tmp,r->∞]

sumH4=Simplify[Sum[H4,{m,-∞,+∞}]]

H=((((((2π m)/β))^2+r^2+v)^4((((2π m)/β))^2+r^2-v)^4)/(((((2π m)/β))^2+r^2)^8));

Hexp=Simplify[Series[H,{v,0,4}]]

H4=SeriesCoefficient[Hexp,4]

sumH4=Simplify[Sum[H4,{m,-∞,+∞}]]

Limit[r^7sumH4/.{r->(x/β)},x->∞]

H=((((((2π m)/β)+(π/β))^2+r^2+v)^4((((2π m)/β)+(π/β))^2+r^2-v)^4)/(((((2π m)/β))^2+r^2)^6((((2π m)/β))^2+r^2+2v)((((2π m)/β))^2+r^2-2v)));

Hexp=Simplify[Series[H,{v,0,4}]]

H=((((((2π m)/β)+(π/β))^2+r^2+λ^2 V+λ v)^4((((2π m)/β)+(π/β))^2+r^2+λ^2 V-λ v)^4)/(((((2π m)/β))^2+r^2+λ^2 V)^6((((2π m)/β))^2+r^2+λ^2 V+2λ v)((((2π m)/β))^2+r^2+λ^2 V-2λ v)));

Hexp=Simplify[Series[H,{λ,0,3}]]

Series[Log[1+x],{x,0,4}]

Product[((m^2+b^2)/(m^2+a^2)),{m,-∞,+∞}]

Product[(((((((π+2 m π))^2)+(r^2) (β^2)))^8)/(((4 (m^2) (π^2)+(r^2) (β^2)))^8)),{m,-∞,+∞}]

H0=SeriesCoefficient[Hexp,0]

H2=SeriesCoefficient[Hexp,2]

H4=SeriesCoefficient[Hexp,4]

sumH2=Sum[H2,{m,-∞,+∞}];

sumH4=Sum[H4,{m,-∞,+∞}];

Chop[(1/(8π^2))(r^5sumH2)/.{r->(10000./β)}]

Chop[(16/15)(r^7sumH4)/.{r->(10000./β)}]

Mat=(({{M, 0, 0, 0, 0, 0, 0, 0, 0, 2 v1}, {0, M, 0, 0, 0, 0, 0, 0, 0, 2 v2}, {0, 0, M, 0, 0, 0, 0, 0, 0, 2v3}, {0, 0, 0, M, 0, 0, 0, 0, 0, 2v4}, {0, 0, 0, 0, M, 0, 0, 0, 0, 2v5}, {0, 0, 0, 0, 0, M, 0, 0, 0, 2v6}, {0, 0, 0, 0, 0, 0, M, 0, 0, 2v7}, {0, 0, 0, 0, 0, 0, 0, M, 0, 2v8}, {0, 0, 0, 0, 0, 0, 0, 0, M, 2v9}, {-2v1, -2v2, -2v3, -2v4, -2v5, -2v6, -2v7, -2v8, -2v9, M}}));

Eigenvalues[Mat]

xvec={x1,x2,x3,x4};

vvec={v1,v2,v3,x4};

M=Table[0,{ii,0,4},{jj,0,4}];

M[[1,1]]=xvec.xvec;

Do[
M[[1,ii+1]]=0;
M[[ii+1,1]]=0;
,{ii,1,4}]

Do[
M[[ii+1,jj+1]]=If[ii==jj,m^2+xvec.xvec,0]-xvec[[ii]]xvec[[jj]];
,{ii,1,4},{jj,1,4}]

MatrixForm[M]

Eigenvalues[M]

M=Table[
If[ii==jj,m^2+xvec.xvec,0]-xvec[[ii]]xvec[[jj]]
,{ii,1,4},{jj,1,4}];

MatrixForm[M]

Eigenvalues[M]

H=Log[((((((2π m)/β)+(π/β))^2+r^2+v)^4((((2π m)/β)+(π/β))^2+r^2-v)^4)/(((((2π m)/β))^2+r^2)^6((((2π m)/β))^2+r^2+2v)((((2π m)/β))^2+r^2-2v)))];

Hexp=Simplify[Series[H,{v,0,4}]]

H0=SeriesCoefficient[Hexp,0]

H2=SeriesCoefficient[Hexp,2]

H4=SeriesCoefficient[Hexp,4]

FullSimplify[(((((2 π m)/R)+a)-I x)/((((2 π m)/R)+b)-I y))(((((2 π m)/R)+a)+I x)/((((2 π m)/R)+b)+I y))]

FullSimplify[Product[(((((2 π m)/R)+a)-I x)/((((2 π m)/R)+b)-I y)),{m,-∞,∞}]Product[(((((2 π m)/R)+a)+I x)/((((2 π m)/R)+b)+I y)),{m,-∞,∞}]]

tmp=Sum[Log[(((2 π m+a)^2+x^2)/((2 π m+b)^2+y^2))]/.{x->0.2,a->0.32,y->0.565,b->0.54},{m,-100,100}]

Log[((Cosh[x]-Cos[a])/(Cosh[y]-Cos[b]))]/.{x->0.2,a->0.32,y->0.565,b->0.54}

H0

Sum[Log[(((((((π+2 m π))^2)+(r^2) (β^2)))^8)/(((4 (m^2) (π^2)+(r^2) (β^2)))^8))]/.{r->0.33,β->0.2},{m,-100,100}]

8Log[((Cosh[r β]-Cos[π])/(Cosh[r β]-Cos[0]))]/.{r->0.33,β->0.2}

Sum[Log[(((((((π+2 m π+a))^2)+(r^2) (β^2)))^8)/((((2 m π+a)^2+(r^2) (β^2)))^8))]/.{a->0.8,r->0.33,β->0.2},{m,-1000,1000}]

8Log[((Cosh[r β]+Cos[a])/(Cosh[r β]-Cos[a]))]/.{a->0.8,r->0.33,β->0.2}

Sum[Log[(((((((π+2 m π+a))^2)+(r^2) (β^2)))^8)/((((2 m π+a)^2+(r^2) (β^2)))^8))]/.{a->0.8,r->5.,β->1},{m,-1000,1000}]

32 Exp[-r β]Cos[a]/.{a->0.8,r->5.,β->1}

Sum[Log[(((((((π+2 m π))^2)+(r^2) (β^2)))^8)/(((4 (m^2) (π^2)+(r^2) (β^2)))^8))]/.{r->5.,β->1},{m,-100000,100000}]

32 Exp[-r β]/.{r->5.,β->1}

tmp=8Log[((Cosh[r β]-Cos[π])/(Cosh[r β]-Cos[0]))]/.{r->-(1/β)Log[(1/x)]};

Series[tmp,{x,0,3}]

tmp=Sum[H2,{m,-∞,∞}]

(1/(β^2))r^2Exp[+r β]tmp/.{β->2.3,r->100000.}

(1/(β^2))r^2Exp[+r β]tmp/.{β->4,r->10000.}

tmp=Sum[H4,{m,-∞,∞}]

(1/β)r^7tmp/.{β->2.3,r->100000.}

(1/β)r^7tmp/.{β->4,r->10000.}

Solve[x^2Exp[x]==β^3,x]

FullSimplify[((Gamma[x])/(Gamma[1+x]))]

Series[Tanh[x],{x,0,3}]

Plot[(2/β) ProductLog[((Sqrt[(β^3)])/2)],{β,0.2,10},PlotRange->All]

(((2/β) ProductLog[((Sqrt[(β^3)])/2)])/((3/β)Log[β]))/.{β->10.^1000}

(((2/β) ProductLog[((Sqrt[(β^3)])/2)])/((2/β)Log[((Sqrt[(β^3)])/2)]))/.{β->10.^1000}

Plot[(((2/β) ProductLog[((Sqrt[(β^3)])/2)])/((3/β)Log[β])),{β,2,100000},PlotRange->All]

Plot[(2/β) ProductLog[((Sqrt[(β^3)])/2)],{β,0.2,100},PlotRange->All]

Plot[{Exp[-x]/.{x->2 ProductLog[((Sqrt[(β^3)])/2)]},(1/(β^(9/5)))},{β,0.2,8}]

Plot[β^(9/5)Exp[-x]/.{x->2 ProductLog[((Sqrt[(β^3)])/2)]},{β,1,1000},PlotRange->All]

Integrate[Exp[-σ a+σ^3 b],{σ,0,∞}]

Simplify[Series[((Tanh[v σ]-(v σ))/(v^3)),{v,0,5}]]

Sact=(1/σ)Sqrt[((2 v)/( σ^(p-1)Sinh[2 v σ]))]Exp[((σ^3)/3)v^2 μdotv^2-(2/15)σ^5v^4 μdotv^2];

PowerExpand[Simplify[Series[Sact,{v,0,4}]]]

tmp=(n-4)+(Exp[2 v σ]+Exp[-2 v σ])-(((n-2))/2)(Exp[v σ]+Exp[-v σ]);

FullSimplify[tmp]

FullSimplify[tmp/.{n->10}]

FullSimplify[(n-4)+(Exp[2 v σ]+Exp[-2 v σ])]

FullSimplify[-(((n-2))/2)(Exp[v σ]+Exp[-v σ])]

ΓBoson=(1/σ)Sqrt[((2 v)/( σ^(p-1)Sinh[2 v σ]))]Exp[((σ^3)/3)v^2 μdotv^2-(2/15)σ^5v^4 μdotv^2]((n-4)+(Exp[2 v σ]+Exp[-2 v σ]));

PowerExpand[FullSimplify[Series[ΓBoson,{v,0,4}]]]

ΓFermion=(1/σ)Sqrt[((2 v)/( σ^(p-1)Sinh[2 v σ]))]Exp[((σ^3)/3)v^2 μdotv^2-(2/15)σ^5v^4 μdotv^2](-(n-2) Cosh[v σ]);

PowerExpand[FullSimplify[Series[ΓFermion,{v,0,4}]]]

4*5/360

2^5

6-2

(((14-4))/3)

((4-2)/6)

((82-11*6)/45)

((7*4)/90)

Integrate[σ^q Exp[-σ μ],{σ,0,∞}]