SetDirectory["~/doc/research/first_order_singularities/paper"];

<<IsingScalingFunction`

data2={θ0->1.14841,θYL->0.989667,CYL->-0.172824,G[1]->-0.310183,G[2]->0.247454,g[0]->0.373691,g[1]->-0.0216363}

prep2={θ0,θYL,ruleB[θ0,{g[0],g[1]}],ruleAL[θ0,{g[0],g[1]}],CYL,{G[1],G[2]},{g[0],g[1]}}/.data2

Plot[Evaluate@Last[(DScriptF0DηList@@prep2)[6,θ]],{θ,0,1.1483}]

Plot[Evaluate[g[θ0,{g[0],g[1]}][θ]/.data2],{θ,0,1.148}]

s

(DufDuh[θ0,θYL,B,C0,CYL,{},{g[0]}])[2][R,θ]

χ0=Limit[(DufDuh[θ0,θYL,B,C0,CYL,{},{g[0],g[1]}])[2][R,θ],θ->θ0,Direction->"FromBelow",Assumptions->{θ0>1,R>0,B>0,θ0>0,θYL>0,g[0]>0,g[1]>0}]

(1/(8 (R^(7/4)) ((g[0])^2)))(-1+((θ0)^2)) (-(1/π)+((2 ((θ0)^2))/(π (-1+((θ0)^2))))+((48 ((θ0)^2) (CYL π (-2 ((θYL)^(5/6))+(((-I θ0+θYL))^(5/6))+(((I θ0+θYL))^(5/6)))+2 C0 (E^(1/(B θ0))) θ0 ExpIntegralEi[-(1/(B θ0))]-2 C0 (E^(1/(2 B θ0))) θ0 ExpIntegralEi[-(1/(2 B θ0))]))/(π (((-1+((θ0)^2)))^3)))-((8 (CYL π (-2 ((θYL)^(5/6))+(((-I θ0+θYL))^(5/6))+(((I θ0+θYL))^(5/6)))+2 C0 (E^(1/(B θ0))) θ0 ExpIntegralEi[-(1/(B θ0))]-2 C0 (E^(1/(2 B θ0))) θ0 ExpIntegralEi[-(1/(2 B θ0))]))/(π (((-1+((θ0)^2)))^2)))-((16 θ0 ((C0/π)-(5/6) I CYL ((1/(((-I θ0+θYL))^(1/6)))-(1/(((I θ0+θYL))^(1/6))))-((C0 (E^(1/(2 B θ0))) (-1+2 B θ0) ExpIntegralEi[-(1/(2 B θ0))])/(2 B π θ0))))/(((-1+((θ0)^2)))^2))+((2 (((2 B C0)/π)+(5/36) CYL ((1/(((-I θ0+θYL))^(7/6)))+(1/(((I θ0+θYL))^(7/6))))+((C0 (2 B θ0 (-1+2 B θ0)-(E^(1/(2 B θ0))) ExpIntegralEi[-(1/(2 B θ0))]))/(8 (B^2) π ((θ0)^3)))))/(-1+((θ0)^2)))-((3 (2+3 ((θ0)^2)) (((θ0 (((-1+((θ0)^2)))^2))/π)+((8 θ0 (CYL π (-2 ((θYL)^(5/6))+(((-I θ0+θYL))^(5/6))+(((I θ0+θYL))^(5/6)))+2 C0 (E^(1/(B θ0))) θ0 ExpIntegralEi[-(1/(B θ0))]-2 C0 (E^(1/(2 B θ0))) θ0 ExpIntegralEi[-(1/(2 B θ0))]))/π)-2 (-1+((θ0)^2)) ((C0/π)-(5/6) I CYL ((1/(((-I θ0+θYL))^(1/6)))-(1/(((I θ0+θYL))^(1/6))))-((C0 (E^(1/(2 B θ0))) (-1+2 B θ0) ExpIntegralEi[-(1/(2 B θ0))])/(2 B π θ0)))))/(2 θ0 (((-1+((θ0)^2)))^3))))+(((R^2) (((-1+((θ0)^2)))^2) Log[R (-1+((θ0)^2))])/(4 π))/.B->ruleB[θ0,{g[0]}]/.C0->ruleAL[θ0,{g[0]}]/.data2

Limit[(DufDut@@prep2)[2][R,θ],θ->θ]

ComplexPlot[(DufDut@@prep2)[θ],{θ,3}]

(uf@@Most[prep2])[1,θ]

Plot[(uf@@Most[prep2])[1,θ],{θ,-1.148,1.148}]

test=DufDut[θ0,θYL,B,C0,CYL,{},{g[0]}][2][R,θ];

Series[test,{θ,0,0},Assumptions->{R>0,θ0>0,g[0]>1}]

Series[DScriptF0DηList[θ0,θYL,B,C0,CYL,{},{g[0]}][2,θ][[-1]],{θ,0,0},Assumptions->{R>0,θ0>0,g[0]>1}]

8/60

16/120

D[96 (B^2) ((θ0)^3) ((θYL)^(7/6)) Log[(θ^2) ((g[0])^2)] (θ^2),{θ,2}]

Series[test,{θ,θ0,0},Assumptions->{R>0,θ0>0,g[0]>1}]

uf[θ0,θYL,B,C0,CYL,{}][R,θ]

Series[(1-(1/((g[θ0,{g[0]}][θ])^(2/Δ)))),{θ,0,1},Assumptions->{θ>0}]

dat=Flatten[Table[{ut[R,θ],uh[θ0,{g[0],g[1]}][R,θ],-(DufDuh@@prep2)[1][R,θ]}/.data2,{R,0.05,2,0.05},{θ,-1.14841,1.14841,0.05}],1];

ListPlot3D[dat]

Plot[Evaluate[(DufDut@@prep2)[2][R,0.1]],{R,0,2}]

ParametricPlot[{θ,Evaluate[(DufDut@@prep2)[2][.1,θ]]},{θ,-1.148,1.148},AspectRatio->1]

dGdξ

(dΦdηList@@prep2)[0,1]

ListLogPlot[Abs[(DScriptFPlusMinusDξθ0List@@prep2)[10]]]

ListLogPlot[Abs[(DScriptF0DηList@@prep2)[10,0.5]]]