ClearAll[b]

b[d_]:=Piecewise[{{7/3,d==3}},(d-3)d/2]

ℋ[d_][X_]:=(X^(-b[d]))Exp[-Sum[A[i]/(X^i),{i,1,d-1}]]f[X]

Series[(X^(-b[d]))Exp[-A/(X^(d-1))]f[0]/(Exp[-A/(Y^(d-1))](Y^(-b[d])))/.X->Y+Sum[(Subscript[a, n])(Y^n),{n,d+1,d+100}]/.d->3,{Y,0,3}]

Series[1/(((x+Sum[(Subscript[X, n])(x^n),{n,d+1,20}]))^(d-1))/.d->3,{x,0,10},Assumptions->{d≥2,d∈Integers}]

SeriesCoefficient[ℋ[d][X[Y]]/Exp[-A/(Y^(d-1))](Y^(b[d]))/.X[Y]->Y+Sum[(Subscript[a, n])(Y^n),{n,d+1,∞}],{Y,0,1}]

D[ℋ[d][X[Y]]/Exp[-A/(Y^(d-1))](Y^(b[d]))/.X[Y]->Y+Sum[(Subscript[a, n])(Y^n),{n,d+1,∞}],{Y,0}]/.{Y->0}//. (0^(_)):>0

Binomial[d-1,1]

Binomial[d,2]

Multinomial[2]

c12=SeriesCoefficient[ℋ[d][X[Y]]/Exp[-A[1]/(Y^(d-1))](Y^(b[d]))/.X[Y]->Y+(Subscript[a, 2])(Y^2)+Sum[(Subscript[a, i])(Y^i),{i,3,d+3}]/.d->2,{Y,0,1}]

c12=SeriesCoefficient[ℋ[d][X[Y]]/Exp[-A[2]/(Y^(d-1))](Y^(b[d]))/.X[Y]->Y+(A[1]/(2A[2]))(Y^2)+(((A[1])^2)/(8((A[2])^2)))(Y^3)+Sum[(Subscript[a, i])(Y^i),{i,4,d+3}]/.d->3,{Y,0,3}]//Expand

c12=SeriesCoefficient[ℋ[d][X[Y]]/Exp[-A[3]/(Y^(d-1))](Y^(b[d]))/.X[Y]->Y+(A[2]/(3A[3]))(Y^2)+(A[1]/(3A[3]))(Y^3)-((((A[2])^3)/(27 ((A[3])^2)))-((A[1] A[2])/(3 A[3])))/(3 A[3])(Y^4)+Sum[(Subscript[a, i])(Y^i),{i,5,d+3}]/.d->4,{Y,0,2}]//Expand

Series[((A[3])/(Y^3))-((A[3])/(((Y+(Y^2) (Subscript[a, 2])+(Y^3) (Subscript[a, 3])+(Y^4) (Subscript[a, 4])+(Y^5) (Subscript[a, 5])+(Y^6) (Subscript[a, 6])+(Y^7) (Subscript[a, 7])))^3))-((A[2])/(((Y+(Y^2) (Subscript[a, 2])+(Y^3) (Subscript[a, 3])+(Y^4) (Subscript[a, 4])+(Y^5) (Subscript[a, 5])+(Y^6) (Subscript[a, 6])+(Y^7) (Subscript[a, 7])))^2))-((A[1])/(Y+(Y^2) (Subscript[a, 2])+(Y^3) (Subscript[a, 3])+(Y^4) (Subscript[a, 4])+(Y^5) (Subscript[a, 5])+(Y^6) (Subscript[a, 6])+(Y^7) (Subscript[a, 7])))/.(Subscript[a, 2])->A[2]/(3A[3]),{Y,0,2}]

s12=Solve[c12==0,(Subscript[a, 3])]

c22=SeriesCoefficient[ℋ[2][X[Y]]/Exp[-A/Y](Y^(b[2]))/.X[Y]->Y+(Subscript[a, 3])(Y^3)+(Subscript[a, 4])(Y^4),{Y,0,2}]

s22=Solve[c22==0/.s12[[1]],(Subscript[a, 4])]

c32=SeriesCoefficient[ℋ[2][X[Y]]/Exp[-A/Y](Y^(b[2]))/.X[Y]->Y+(Subscript[a, 3])(Y^3)+(Subscript[a, 4])(Y^4)+(Subscript[a, 5])(Y^5)+(Subscript[a, 6])(Y^6)+(Subscript[a, 7])(Y^7),{Y,0,3}]

s32=Solve[c32==0/.s12[[1]]/.s22[[1]],(Subscript[a, 5])]

c42=SeriesCoefficient[ℋ[2][X[Y]]/Exp[-A/Y](Y^(b[2]))/.X[Y]->Y+(Subscript[a, 3])(Y^3)+(Subscript[a, 4])(Y^4)+(Subscript[a, 5])(Y^5)+(Subscript[a, 6])(Y^6)+(Subscript[a, 7])(Y^7),{Y,0,4}]

Solve[c42==0/.s12[[1]]/.s22[[1]]/.s32[[1]],(Subscript[a, 6])]

c13=SeriesCoefficient[ℋ[3][X[Y]]/Exp[-A/(Y^2)](Y^(b[3]))/.X[Y]->Y+(Subscript[a, 4])(Y^4)+(Subscript[a, 5])(Y^5)+(Subscript[a, 6])(Y^6)+(Subscript[a, 7])(Y^7),{Y,0,1}]

s13=Solve[c13==0,(Subscript[a, 4])]

c23=SeriesCoefficient[ℋ[3][X[Y]]/Exp[-A/(Y^2)](Y^(b[3]))/.X[Y]->Y+(Subscript[a, 4])(Y^4)+(Subscript[a, 5])(Y^5)+(Subscript[a, 6])(Y^6)+(Subscript[a, 7])(Y^7),{Y,0,2}]

s23=Solve[c23==0/.s13[[1]],(Subscript[a, 5])]

c33=SeriesCoefficient[ℋ[3][X[Y]]/Exp[-A/(Y^2)](Y^(b[3]))/.X[Y]->Y+(Subscript[a, 4])(Y^4)+(Subscript[a, 5])(Y^5)+(Subscript[a, 6])(Y^6)+(Subscript[a, 7])(Y^7),{Y,0,3}]

s33=Solve[c33==0/.s13[[1]]/.s23[[1]],(Subscript[a, 6])]

c43=SeriesCoefficient[ℋ[3][X[Y]]/Exp[-A/(Y^2)](Y^(b[3]))/.X[Y]->Y+(Subscript[a, 4])(Y^4)+(Subscript[a, 5])(Y^5)+(Subscript[a, 6])(Y^6)+(Subscript[a, 7])(Y^7),{Y,0,4}]

Solve[c43==0/.s13[[1]]/.s23[[1]]/.s33[[1]],(Subscript[a, 7])]

c14=SeriesCoefficient[ℋ[4][X[Y]]/Exp[-A/(Y^3)](Y^(b[4]))/.X[Y]->Y+(Subscript[a, 5])(Y^5)+(Subscript[a, 6])(Y^6)+(Subscript[a, 7])(Y^7)+(Subscript[a, 8])(Y^8),{Y,0,1}]

s14=Solve[c14==0,(Subscript[a, 5])]

c24=SeriesCoefficient[ℋ[4][X[Y]]/Exp[-A/(Y^3)](Y^(b[4]))/.X[Y]->Y+(Subscript[a, 5])(Y^5)+(Subscript[a, 6])(Y^6)+(Subscript[a, 7])(Y^7)+(Subscript[a, 8])(Y^8),{Y,0,2}]

s24=Solve[c24==0/.s14[[1]],(Subscript[a, 6])]

c34=SeriesCoefficient[ℋ[4][X[Y]]/Exp[-A/(Y^3)](Y^(b[4]))/.X[Y]->Y+(Subscript[a, 5])(Y^5)+(Subscript[a, 6])(Y^6)+(Subscript[a, 7])(Y^7)+(Subscript[a, 8])(Y^8),{Y,0,3}]

s34=Solve[c34==0/.s14[[1]]/.s24[[1]],(Subscript[a, 7])]

c44=SeriesCoefficient[ℋ[4][X[Y]]/Exp[-A/(Y^3)](Y^(b[4]))/.X[Y]->Y+(Subscript[a, 5])(Y^5)+(Subscript[a, 6])(Y^6)+(Subscript[a, 7])(Y^7)+(Subscript[a, 8])(Y^8),{Y,0,4}]

Solve[c44==0/.s14[[1]]/.s24[[1]]/.s34[[1]],(Subscript[a, 8])]

s22

s23

s24

s32

s33

s34

Binomial[k,n]Binomial[n,j]//FullSimplify

Binomial[2,1]

SeriesCoefficient[Exp[f[x]],{x,0,4}]

Binomial[d-2+1,1](((-1))^1)Multinomial[1,0,0,0,0,0]

Series[(((x f[x]))^(-7/3))/(((x))^(-7/3)),{x,0,2}]/.f[0]->1//Simplify

Series[(((x f[x]))^(-b))/(((x))^(-b)),{x,0,2}]/.f[0]->1/.b->7/3//Simplify

Binomial[-7/3,1]

f@@Range[3]

y[d_][m_]:=With[{js=Tally/@IntegerPartitions[m,All,Range[m]]},
Total[With[{k=Total[#[[All,2]]]},
Binomial[d-2+k,k](((-1))^k)Multinomial@@#[[All,2]]Times@@(((Subscript[a, #1+1])^(#2))&@@@#)
]&/@js]
]

z[d_][n_]:=With[{js=Tally/@IntegerPartitions[n,All,Range[n]]},
 Total[With[{k=Total[#[[All,2]]]},
((((-(Subscript[A, d-1])))^k)/(k!))Multinomial@@#[[All,2]]Times@@(((y[d][#1+d-1])^(#2))&@@@#)
]&/@js]
]

w[d_][m_]:=With[{js=Tally/@IntegerPartitions[m,All,Range[m]]},
Total[With[{k=Total[#[[All,2]]]},
Binomial[-b[d],k]Multinomial@@#[[All,2]]Times@@(((Subscript[a, #1+1])^(#2))&@@@#)
]&/@js]
]

v[d_][m_]:=Sum[z[d][i]w[d][m-i],{i,0,m}]

v[2][1]

ReleaseHold[Hold[Sum[z[d][m-i]w[d][i],{i,d,m}]]/.{d->3,m->3}]

w[3][3]

v[3][3]

ClearAll[r]

r[d_][k_]:=First@Solve[0==y[d][d-1],(Subscript[a, d])]/;k==d-1

r[4][1]//Simplify

r[d_][k_]:=Thread[Rule[#[[All,1]],(#[[All,2]]/.#)]]&@(Join[First@Solve[(Subscript[A, d-1-k])/(Subscript[A, d-1])==y[d][k],(Subscript[a, 1+k])],#]&@r[d][k+1])

s[d_][0]:=r[d][1]

s[d_][k_]:=Join[First@Solve[v[d][k]==(Subscript[B, k])/.#,(Subscript[a, d+k])],#]&@s[d][k-1]

s[2][4]//Simplify

test=With[{ss=s[2][10]},((Subscript[a, 1+#])/.ss/.(Subscript[B, j_]):>(c^(j+1))/.(Subscript[A, _]):>1/c)&/@Range[10]];

Show[ListLogPlot[Abs[test/.c->10]],LogPlot[ (10^(2k)),{k,0,20}]]

coeff[d_][k_]:=With[{ss=s[d][k]},CoefficientList[(Subscript[a, d+#])/.ss/.(Subscript[B, j_]):>(c^(j+1))/.A->1/c,{c}]&/@Range[k]]

cc=coeff[2][14];

ListPlot[Max/@(cc/.b[2]->-1)]

data={2+#1,Length[#2]}&@@@Thread[{Range[Length[cc]],cc}];

Show[ListPlot[data],Plot[2x,{x,0,16}]]

cc3=coeff[3][13];

cc3/

data={3+#1,Length[#2]}&@@@Thread[{Range[Length[cc3]],cc3}];

Show[ListPlot[data],Plot[2x,{x,0,16}]]

coeff[3][8]

g[d_][n_]:=v[d][n]-A(d-1)(Subscript[a, d+n])

g[2][5]/.b[2]->-1//Simplify

t1=v[3][8]//Simplify

D[Y+Sum[(Subscript[a, n])(Y^n),{n,d+1,d+100}]/.d->2,{Y,4}]/4!/.Y->0

t2=SeriesCoefficient[(X^(-b[d]))Exp[-(Subscript[A, d-1])/(X^(d-1))]/(Exp[-Sum[(Subscript[A, i])/(Y^i),{i,1,d-1}]](Y^(-b[d])))/.X->Y-(Subscript[A, 1])/(2 (Subscript[A, 2]))(Y^2)+(3/8)(((Subscript[A, 1])^2)/((Subscript[A, 2])^2))(Y^3)+Sum[(Subscript[a, n])(Y^n),{n,4,d+5}]/.d->3,{Y,0,1}]

Binomial[d+1,3]

Multinomial[1,1]

Series[((Subscript[A, 2])/(Y^2))-((Subscript[A, 2])/(((Y+(Y^2) (-(Subscript[A, 1])/(2 (Subscript[A, 2])))+(Y^3) (Subscript[a, 3])+(Y^4) (Subscript[a, 4])+(Y^5) (Subscript[a, 5])+(Y^6) (Subscript[a, 6])+(Y^7) (Subscript[a, 7])+(Y^8) (Subscript[a, 8])))^2)),{Y,0,1}]

t1==t2

y[d_][m_]:=

(* Constants fixed for the 2D square-lattice Ising model *)
Δ:=15/8
γ:=7/4
β:=1/8
ℳ0:=((((2^(5/2))ArcSinh[1]))^β)
Tc:=2/Log[1+Sqrt[2]]
(Subscript[A, 1]):=(Tc^2)ℳ0/(16 π)/((((((2ArcSinh[1]))^2)(Ch (((2ArcSinh[1]))^(-Δ)))))^(-1))

ClearAll[Subscript]

(* Constants from Mangazeev et al *)
Ch:=0.838677624411
Ghs={0,0,-1.8452280782328,0,8.333711750,0,-95.16896,0,1457.62,0,-25891};
Gls:={0,-1.3578383417066,-0.048953289720,0.038863932,-0.068362119,0.18388370,-0.6591714,2.937665,-15.61,96.76,-6.79 (10^2),5.34 (10^3),-4.66 (10^4),4.46 (10^5),-4.66 (10^6)};
Φs:={-1.197733383797993,-0.318810124891,0.110886196683,0.01642689465,-2.639978 (10^(-4)),-5.140526 (10^(-4)),2.08856 (10^(-4)),-4.4819 (10^(-5))};

(* Translation between the variables of Mangazeev et al and our own *)
G2C[G_,n_]:=(((2ArcSinh[1]))^2)(((Ch (((2ArcSinh[1]))^(-Δ))))^n)G
Gs2Cs[Gs_]:=MapIndexed[G2C[#1,#2[[1]]-1]&,Gs]
Φ2C[Φ_,n_]:=(Ch^(2/Δ))(((-2ArcSinh[1](Ch^(-1/Δ))))^n)Φ
Φs2Cs[Gs_]:=MapIndexed[Φ2C[#1,#2[[1]]-1]&,Gs]

ClearAll[B]

cs=Gls

ℱseries[d_][n_]:=(((((-1))^(1+n)) ((Subscript[A, d-1])^(-((n+b[d])/(d-1)))) Gamma[((n+b[d])/(d-1))] (Subscript[B, 0]))/((d-1) π))/;n+b[d]>0

Show[ListLogPlot[Abs@{cs-ℱseries[2]/@(Range[15]-1)/.s1/.(Subscript[A, 1])->1.15}]]

Plot[x(1+LaguerreL[#,x]Exp[-x/2])&/@Range[10],{x,0,5}]

ℱseries[2][1]:=Gls[[2]]

N[ℳ0]

SeriesCoefficient[f[x],{x,0,10}]10!

(Subscript[A, 1]):=1.15

g[x_]:=x(1+Sum[(Subscript[a, 1+i])(x^i),{i,2,12}])

ClearAll[f]

s1=First@Solve[cs[[1+2]]==SeriesCoefficient[f[g[x]],{x,0,2}]/.Derivative[n_][f][0]:>ℱseries[2][n]n!,(Subscript[B, 0])];

s1

s2=First@Solve[cs[[1+3]]==SeriesCoefficient[f[g[x]],{x,0,3}]/.Derivative[n_][f][0]:>ℱseries[2][n]n!/.s1,(Subscript[a, 3])];

s2

s3=First@Solve[cs[[1+4]]==SeriesCoefficient[f[g[x]],{x,0,4}]/.Derivative[n_][f][0]:>ℱseries[2][n]n!/.Join[s1,s2],(Subscript[a, 4])];

s3

s4=First@Solve[cs[[1+5]]==SeriesCoefficient[f[g[x]],{x,0,5}]/.Derivative[n_][f][0]:>ℱseries[2][n]n!/.Join[s1,s2,s3],(Subscript[a, 5])];

s4

s5=First@Solve[cs[[1+6]]==SeriesCoefficient[f[g[x]],{x,0,6}]/.Derivative[n_][f][0]:>ℱseries[2][n]n!/.Join[s1,s2,s3,s4],(Subscript[a, 6])];

s5

s6=First@Solve[cs[[1+7]]==SeriesCoefficient[f[g[x]],{x,0,7}]/.Derivative[n_][f][0]:>ℱseries[2][n]n!/.Join[s1,s2,s3,s4,s5],(Subscript[a, 7])];

s6

s7=First@Solve[cs[[1+8]]==SeriesCoefficient[f[g[x]],{x,0,8}]/.Derivative[n_][f][0]:>ℱseries[2][n]n!/.Join[s1,s2,s3,s4,s5,s6],(Subscript[a, 8])];

s7

s8=First@Solve[cs[[1+9]]==SeriesCoefficient[f[g[x]],{x,0,9}]/.Derivative[n_][f][0]:>ℱseries[2][n]n!/.Join[s1,s2,s3,s4,s5,s6,s7],(Subscript[a, 9])];

s8

s9=First@Solve[cs[[1+10]]==SeriesCoefficient[f[g[x]],{x,0,10}]/.Derivative[n_][f][0]:>ℱseries[2][n]n!/.Join[s1,s2,s3,s4,s5,s6,s7,s8],(Subscript[a, 10])];

s9

s10=First@Solve[cs[[1+11]]==SeriesCoefficient[f[g[x]],{x,0,11}]/.Derivative[n_][f][0]:>ℱseries[2][n]n!/.Join[s1,s2,s3,s4,s5,s6,s7,s8,s9],(Subscript[a, 11])];

s11=First@Solve[cs[[1+12]]==SeriesCoefficient[f[g[x]],{x,0,12}]/.Derivative[n_][f][0]:>ℱseries[2][n]n!/.Join[s1,s2,s3,s4,s5,s6,s7,s8,s9,s10],(Subscript[a, 12])];

s12=First@Solve[cs[[1+13]]==SeriesCoefficient[f[g[x]],{x,0,13}]/.Derivative[n_][f][0]:>ℱseries[2][n]n!/.Join[s1,s2,s3,s4,s5,s6,s7,s8,s9,s10,s11],(Subscript[a, 13])];

f=((y^(-b[2]))Exp[-(Subscript[A, 1])/(y^1)](Subscript[B, 0])/.y->g[x])/.Join[s1,s2,s3,s4,s5,s6,s7,s8,s9,s10,s11,s12]

Plot[Evaluate[g[x]/.Join[s1,s2,s3,s4,s5,s6,s7,s8,s9,s10,s11,s12]],{x,0,.5}]

Gls[[2]]

ListLogPlot[Abs@Join[{a->Gls[[2]]},s1,s2,s3,s4,s5,s6,s7,s8,s9,s10,s11,s12][[All,2]]/.(Subscript[A, 1])->.01]

SeriesCoefficient[f[x],{x,0,10}]

ℱseries[3][2]

ClearAll[a]

a=(((Subscript[B, 0])(((-1))^(n+1)))/π)Integrate[(((X))^(-b[d]))Exp[-A/(X^(d-1))]/(X^(n+1)),{X,0,∞},Assumptions->{n+b[d]>0,A>0,d>1}]

a/.b[d]->-1

ClearAll[b]

Solve[0==D[(X^(-b))Exp[-A/(X^(d-1))],X],X]

s=Solve[0==D[X/(1+b (X^3))Exp[-A(1/(X^1))],X],X]

Solve[(X/.s[[1]])==A,b]

-b-d-(-1-b)

(b/(A(d-1)))==  (X^(-(d-1)))

((((b/(A(d-1)))))^(-1/(d-1)))

(E^((-I π-Log[b]+Log[A-A d])/(-1+d)))/.b->2/.d->4//FullSimplify

Simplify[((-I π-Log[b[d]]-Log[(Subscript[B, 0])]+Log[(Subscript[A, -1+d]) (Subscript[B, 0])-d (Subscript[A, -1+d]) (Subscript[B, 0])])/(-1+d)),{(Subscript[B, 0])>0,(Subscript[A, d-1])>0,d>1}]

-((Log[((b[d])/((-1+d) (Subscript[A, -1+d])))])/(-1+d))/.{d->4,b[d]->2,(Subscript[A, d-1])->1}

((((b/(A(d-1)))))^(-1/(d-1)))/.{d->3,b->7/3,A->1}

((((b/(A(d-1)))))^(-1/(d-1)))/.{d->2,b->-1}

Plot[(X^1)Exp[-1/(X^1)],{X,-5,5},GridLines->{{((((b/(A(d-1)))))^(-1/(d-1)))/.{d->2,b->-1,A->1}},None}]

Plot[Evaluate[{X Exp[-A/X],(X^1)Exp[-A(1/(X^1)+2/(A^2) X)],X/(1+1(X^3))Exp[-A/X]}/.A->1],{X,0,10},PlotRange->{0,0.5}]

Integrate[((Xp/(1+(Xp^b)) Exp[-A/Xp])/(Xp+X)),{Xp,0,∞},Assumptions->{X>0,A>0,b>2}]

Plot[Evaluate[(((E^(-I A)) (π+(E^(2 I A)) π-I π X+I (E^(2 I A)) π X+(E^(2 I A)) (-I+X) ExpIntegralEi[-I A]+(I+X) ExpIntegralEi[I A]-2 (E^(A (I+(1/X)))) X ExpIntegralEi[-(A/X)]))/(2 (1+(X^2))))/.A->2],{X,0,10}]

ClearAll[g]

ClearAll[f]

f[x]=g[y[x]]

Let yp=y(xp). Then dyp = y’(xp) dxp, xp=(y^(-1))(yp)

(1/π)(∫_(-∞)^0)((Im[g[xp]])/(y-xp))\[DifferentialD]xp

(1/π)(∫_(-∞)^0)((Im[g[yp]])/(x-(y^(-1))[yp]))(1/(y'[(y^(-1))[yp]]))\[DifferentialD]yp

Series[((Im[g[y]])/(x-y h[y]))(D[y[

Γ[m_,h_]:=((σ h)/(2 π))Exp[-((π (m^2))/(2 σ h))]

Series[Γ[((f[h])^2),((g[h])^2)h/(2 σ)]/Exp[-π ((f[0])^4)/(((g[0])^2)h)],{h,0,3}]

int[n_,m_]:=Integrate[(((x^n) Exp[-1/x])/(((x+y))^m)),{x,0,∞},Assumptions->{y>0}]

Integrate[(x^2),x,x,x]

test=Simplify[A int[1,3]+B Integrate[int[2,4],y]]

test2=Simplify[Integrate[-A int[1,4]+B int[2,4],y]]

Plot[Evaluate[{test,test2}/.{A->1,B->0.1,CC->-0,DD->0}],{y,0,100}]

Integrate[int[2],y]

Integrate[Integrate[x2,y],y]