p1=Plot[I,{x,-1,1},AxesStyle->Dashed,Ticks->False,PlotStyle->{Black,Thick},AxesLabel->{Re[ξ],Im[ξ]},Epilog->{{Thick,Line[{{-1,0},{0,0}}]},Disk[{0,0},0.025]},AspectRatio->1/GoldenRatio,PlotRange->{{-1,1},{-1,1}/GoldenRatio},LabelStyle->{FontFamily->"Times",FontSize->14,Black},ImageSize->500]

Export["~/doc/research/first_order_singularities/paper/figs/F_lower_singularities.eps",p1];

p2=Plot[I,{x,-1,1},AxesStyle->Dashed,Ticks->False,PlotStyle->{Black,Thick},AxesLabel->{Re[ξ],Im[ξ]},Epilog->{{Thick,Line[{{0,1/GoldenRatio/2},{0,1}}],Line[{{0,-1/GoldenRatio/2},{0,-1}}]},RegularPolygon[{0,1/GoldenRatio/2},0.025,4],RegularPolygon[{0,-1/GoldenRatio/2},0.025,4],
Line[{{0,1/GoldenRatio/2},{0.05,1/GoldenRatio/2}}],
{Black,Text[Style["i(Subscript[ξ, YL])",FontSize->14,Black,FontFamily->"Times"], {0.125,1/GoldenRatio/2}]}
},AspectRatio->1/GoldenRatio,PlotRange->{{-1,1},{-1,1}/GoldenRatio},LabelStyle->{FontFamily->"Times",FontSize->14,Black},ImageSize->500]

Export["~/doc/research/first_order_singularities/paper/figs/F_higher_singularities.eps",p2];

p3=Plot[I,{x,-1,1},AxesStyle->Dashed,Ticks->False,PlotStyle->{Black,Thick},AxesLabel->{Re[θ],Im[θ]},Epilog->{{Thick,Line[{{0,1/GoldenRatio/2},{0,1}}],Line[{{0,-1/GoldenRatio/2},{0,-1}}]},RegularPolygon[{0,1/GoldenRatio/2},0.025,4],RegularPolygon[{0,-1/GoldenRatio/2},0.025,4],{Thick,Line[{{-1,0},{-0.75,0}}]},Disk[{-0.75,0},0.025],{Thick,Line[{{1,0},{0.75,0}}]},Disk[{0.75,0},0.025],
Line[{{0.75,0},{0.75,0.05}}],
Line[{{0,1/GoldenRatio/2},{0.05,1/GoldenRatio/2}}],
{Black,Text[Style["i(Subscript[θ, YL])",FontSize->14,Black,FontFamily->"Times"], {0.125,1/GoldenRatio/2}]},
{Black,Text[Style["(Subscript[θ, c])",FontSize->14,Black,FontFamily->"Times"], {0.75,0.1}]}
},AspectRatio->1/GoldenRatio,PlotRange->{{-1,1},{-1,1}/GoldenRatio},LabelStyle->{FontFamily->"Times",FontSize->14,Black},ImageSize->500]

Export["~/doc/research/first_order_singularities/paper/figs/F_theta_singularities.eps",p3];

p4=Plot[(Sqrt[(0.95^2)-(x^2)])/GoldenRatio,{x,-1,1},AxesStyle->Dashed,Ticks->False,PlotStyle->{Red},AxesLabel->{Re[θ'],Im[θ']},Prolog->{
{Thickness[0.004],Red,Line[{{-0.95,0.0075},{-0.8,0.0075}}]},
{Thickness[0.004],Red,Circle[{-0.75,0},0.05,{0,π}]},
{Thickness[0.004],Red,Line[{{0.8,0.0075},{0.95,0.0075}}]},
{Thickness[0.004],Red,Circle[{0.75,0},0.05,{0,π}]},
{Thickness[0.004],Red,Line[{{-0.7,0},{0.-0.05,0}}]},
{Thickness[0.004],Red,Circle[{0,0},0.05,{0,π}]},
{Thickness[0.004],Red,Line[{{0.05,0},{0.25,0}}]},
{Thickness[0.004],Red,Circle[{0.3,0},0.05,{0,π}]},
{Thickness[0.004],Red,Line[{{0.35,0},{0.7,0}}]},
{Thickness[0.004],Red,Line[{{0.0075,1/GoldenRatio 0.95},{0.0075,1/GoldenRatio/2+0.05}}]},
{Thickness[0.004],Red,Line[{{-0.0075,1/GoldenRatio 0.95},{-0.0075,1/GoldenRatio/2+0.05}}]},
{Thickness[0.004],Red,Circle[{0,1/GoldenRatio/2},0.05]},
Line[{{0.3,0},{0.3,-0.05}}],
{Black,Text[Style["θ",FontSize->14,Black,FontFamily->"Times"], {0.3,-0.1}]}},
Epilog->{
{Thick,Line[{{0,1/GoldenRatio/2},{0,1}}],Line[{{0,-1/GoldenRatio/2},{0,-1}}]},RegularPolygon[{0,1/GoldenRatio/2},0.025,4],
Rotate[RegularPolygon[{0,0},0.025,4],π/4],RegularPolygon[{0,-1/GoldenRatio/2},0.025,4],{Thick,Line[{{-1,0},{-0.75,0}}]},Disk[{-0.75,0},0.025],{Thick,Line[{{1,0},{0.75,0}}]},Disk[{0.75,0},0.025],
Polygon[{0.3,0}+#&/@SortBy[0.025/1.5Join[CirclePoints[5],-CirclePoints[5]/2],ArcTan@@#&]]
},AspectRatio->1/GoldenRatio,PlotRange->{{-1,1},{-1,1}/GoldenRatio},LabelStyle->{FontFamily->"Times",FontSize->14,Black},ImageSize->500]

Export["~/doc/research/first_order_singularities/paper/figs/contour_path.eps",p4];

ℐ2[θc_,B_][θ_]:=(θ-θc)Exp[((-1)/(B(θ-θc)))]
ℛ2[θc_,B_][θ_]:=(1/π)(θc Exp[(1/(B θc))]ExpIntegralEi[((-1)/(B θc))]+(θ-θc)Exp[((-1)/(B(θ-θc)))]ExpIntegralEi[(1/(B(θ-θc)))])
ℱc2[θc_,B_,Fc_][θ_]:=Fc(ℛ2[θc,B][θ]+ℛ2[θc,B][-θ]+I Sign[Im[θ]](ℐ2[θc,B][θ]-ℐ2[θc,B][-θ]))
ℱc2n[θc_,B_,Fc_][θ_]:=ℱc2[θc,B,Fc][θ]/;NumericQ[θ]

ℱc3[θc_,B_,Fc_][θ_]:=Fc(((((θc)^2)-(θ^2)))^(-7/3))Exp[-1/(((B(((θc)^2)-(θ^2))))^2)]

Plot[Evaluate[ReIm@ℱc2[1+2/10,1,1][θ+(10^(-10))I]],{θ,-2,2},WorkingPrecision->30,PlotPoints->100]

ComplexPlot[ℱc2n[1.2,1,1][θ],{θ,-4-4I,4+4I},ColorFunction->None,Exclusions->{{Im[θ]==0,((Re[θ])^2)>(1.2^2)}}]

ComplexContourPlot[ReIm@ℱc2n[1.2,1,1][θ],{θ,-2-2I,2+2I},ColorFunction->None,Exclusions->{{Im[θ]==0,((Re[θ])^2)>(1.2^2)}},Contours->30]

Plot[Evaluate[ReIm@ℱc3[1,1,1][θ-(10^(-10))I]],{θ,-2,2},WorkingPrecision->20,PlotPoints->100]

ComplexPlot[ℱc3[1.2,1,1][θ]-11((((0.18^2)+(θ^2)))^(1+0.085)),{θ,-2-2I,2+2I},ColorFunction->None,Exclusions->{{Im[θ]==0,((Re[θ])^2)>(1.2^2)},{Re[θ]==0,((Im[θ])^2)>(0.18^2)}},PlotPoints->50]

Series[(((((θc)^2)-(θ^2)))^(-7/3)),{θ,θc,1}]