Type (1,2,n)+(0,0,1)

This notebook shows spheres with an end of type (1, 2, n) and an end of type (0, 0, 1)

Data

ω1[z_] :=1/z;
ω2[z_] :=1/z^2;
ω3[z_] :=1/z^n+I/z^2;

f[r_,t_]=FullSimplify[ComplexExpand[Re[Integrate[{ω1[z],ω2[z],ω3[z]},z]/.z->r E^(I t)],TargetFunctions->{Re,Im}],0<t<Pi&&r>0]

n=3;

ParametricPlot3D[f[r,t],{r,.4,20},{t,0,2 Pi},PlotRange->All, AxesLabel->{x,y,z}]

Embeddednes and Regularity

f[r,t]

f[r,t1]-f[r,t2]//Factor

Worst case is t2 == -t1 :

%/.t2->-t1//Factor

Embeddedness ok. Regularity:

Cross@@Transpose[D[f[r,t],{{r,t}}]]

cross={-(((Cos[t])^2)/(r^3))-(r^(-1-n)) Cos[t-n t] Sin[t]-(((Sin[t])^2)/(r^3))-(r^(-1-n)) Cos[t] Sin[t-n t],((Cos[t])/(r^2))+(r^(-n)) Sin[t-n t],((Sin[t])/(r^2))}

cross/.t->0//Factor

cross/.t->Pi//Factor

Simplify[%,n∈Integers]