Zeros of the Hankel function with complex parameter

Question

I'm trying to find the zeros of the Hankel function (the first few will do) of the first kind $H^{(1)}_\nu(z) = J_\nu(z) + i Y_\nu(z)$ for complex argument $z$ but I'm not sure what is the best function for this in mathematica.

EDIT: Per suggestion I have tried to implement the function FindAllCrossings2D

Options[FindAllCrossings2D] = 
  Sort[Join[
    Options[FindRoot], {MaxRecursion -> Automatic, 
     PerformanceGoal :> $PerformanceGoal, PlotPoints -> Automatic}]];

FindAllCrossings2D[funcs_, {x_, xmin_, xmax_}, {y_, ymin_, ymax_}, 
  opts___] := 
 Module[{contourData, seeds, tt, 
   fy = Compile[{x, y}, Evaluate[funcs[[2]]]]}, 
  contourData = 
   Map[First, 
    Cases[Normal[
      ContourPlot[funcs[[1]], {x, xmin, xmax}, {y, ymin, ymax}, 
       Contours -> {0}, ContourShading -> False, 
       PlotRange -> {Full, Full, Automatic}, 
       Evaluate[
        Sequence @@ 
         FilterRules[Join[{opts}, Options[FindAllCrossings2D]], 
          DeleteCases[Options[ContourPlot], Method -> _]]]]], _Line, 
     Infinity]];
  seeds = 
   Flatten[Map[#[[1 + 
         Flatten[Position[
           Rest[tt = Sign[Apply[fy, #, 2]]] Most[tt], -1]]]] &, 
     contourData], 1];
  If[seeds == {}, seeds, 
   Select[Union[
     Map[{x, y} /. 
        FindRoot[{funcs[[1]] == 0, 
          funcs[[2]] == 0}, {x, #[[1]]}, {y, #[[2]]}, 
         Evaluate[
          Sequence @@ 
           FilterRules[Join[{opts}, Options[FindAllCrossings2D]], 
            Options[FindRoot]]]] &, 
      seeds]], (xmin < #[[1]] < xmax && ymin < #[[2]] < ymax) &]]]

sols = FindAllCrossings2D[{Re[HankelH1[0, x + I y]], 
   Im[HankelH1[0, x + I y]]}, {x, -2, 2}, {y, -2, 2}]

But I receive this error "FindRoot::cvmit: Failed to converge to the requested accuracy or precision within 100 iterations."

I also looked at the contour plot and we can clearly see nontrivial zeros periodically for z = $x + i y$ and $x < 0$ and $y < 0$. I also attempted the get coordinate and then FindRoot[], but this failed to converge and appeared to be going away from the root.

Answer 1

You can use Solve for this purpose, as long as you restrict the domain:

zeros = z /. Solve[HankelH1[0, z] == 0 && -10 < Re[z] < 10 && -10 < Im[z] < 10, z]

Solve::incs: Warning: Solve was unable to prove that the solution set found is complete.

{Root[{HankelH1[ 0, #1] &, -8.65370576584112448841350498548699240993120635923782334125675 - 0.34600819276767292735146122876446089332681742821336233794111 I}], Root[{HankelH1[ 0, #1] &, -5.519997520841832519785940731628753302839051099087900948702 - 0.345225028545679355074185641966127944569911027495087086248 I}], Root[{HankelH1[ 0, #1] &, -2.4040911771553443579757061017638843409511660145346662653424 - 0.3405021529561410696628419946032815758353900295131219830737 I}]}

Unfortunately, Solve doesn't always find the complete solution set. Let's check whether the above roots are actually zeros:

HankelH1[0, N[zeros, 100]]

{0.*10^-101 + 0.*10^-101 I, 0.*10^-98 + 0.*10^-98 I, 0.*10^-99 + 0.*10^-99 I}