f1[x_]:=-x Exp[1/x]

f2[x_]:=(1/π)x Exp[1/x]ExpIntegralEi[-1/x]

f3[x_]:=f2[x]+I Sign[Im[x]]f1[x]

Integrate[2((θ^2)/π)

FullSimplify[{(((z Exp[I π/(2 Δ)]))^(-Δ)),(z^(-Δ))}/.{Δ->RandomReal[],z->RandomReal[]},Assumptions->{z>0,Δ>0}]

(

Integrate[((ϕ Exp[-1/ϕ](((1+((((ϕ+θc)/ϕ0))^2)))^(5/6)))/((((ϕ+θc))^3)(ϕ+θc-θ))),{ϕ,0,∞},Assumptions->{θc>0,B>0,θ<θc,θ>0,ϕ0>0}]

ComplexContourPlot[ReIm[f1[x+1.2]+f1[-x-1.2]],{x,

f3[x+I 0.001]/.x->-1

D[ComplexExpand[Re[Simplify[f3[x+I y],Assumptions->{x>0,y>0}]],TargetFunctions->Conjugate],x]/.y->0//Simplify

D[ComplexExpand[Re[Simplify[f3[x+I y],Assumptions->{x>0,y<0}]],TargetFunctions->Conjugate],x]/.y->0//Simplify

D[ComplexExpand[Im[Simplify[f3[1+I y],Assumptions->{x>0,y>0}]],TargetFunctions->Conjugate],y]/.y->0//Simplify

D[ComplexExpand[Im[Simplify[f3[1+I y],Assumptions->{x>0,y<0}]],TargetFunctions->Conjugate],y]/.y->0//Simplify

D[ComplexExpand[Re[Simplify[f3[1+I y],Assumptions->{x>0,y>0}]],TargetFunctions->Conjugate],y]/.y->0//Simplify

D[ComplexExpand[Re[Simplify[f3[1+I y],Assumptions->{x>0,y<0}]],TargetFunctions->Conjugate],y]/.y->0//Simplify

D[ComplexExpand[Im[Simplify[f3[1+I y],Assumptions->{x>0,y>0}]],TargetFunctions->Conjugate],x]/.y->0//Simplify

Limit[Simplify[D[Simplify[ComplexExpand[Im[Simplify[f3[1+I y],Assumptions->{x>0,y<0}]],TargetFunctions->Conjugate],Assumptions->{x>0,y<0}],x],Assumptions->{x>0,y<0}],y->0,Direction->"FromAbove"]

d1=D[Simplify[f3[z],Assumptions->{Im[z]>0}],z]

d2=D[Simplify[f3[z],Assumptions->{Im[z]<0}],z]

Plot[Im[{d1,d2}/.z->2+I y],{y,-1,1}]

D[ComplexExpand[Im[Simplify[f3[x+I y],Assumptions->{x>0,y>0}]]],y]/.y->0

Simplify[f3[x+I y],Assumptions->{x>0,y>0}]

ComplexExpand[D[f3[x+I y],{{x,y}}]]//Simplify

ComplexExpand[f3[z]]

Plot[Im@f3[x+0.00001I ],{x,-2,2}]

f3[0.5]

Sign[0]

ComplexContourPlot[ReIm[f3[x]],{x,-3-3I,3+3I},Exclusions->None,Contours->40]