How to show that $f(z) = z^2 + 1$ intersects the line through the roots of $f(z) = 0$ in ComplexPlot3D?

Question

I have plotted the complex function $f(z) = z^2+1$ by using the following code:

ComplexPlot3D[z^2+1, {z, -5 - 5 I, 5 + 5 I}]

However, I want to show that $z^2+1$ intersects the line $\Re(z) = 0$. The z-coordinates of intersection are: $z = i$ and $z = -i$, because the roots can easily be found by factorising the function: $f(z) = (z+i)(z-i)$. How do I show that the function $f(z)$ intersects the line $\Re(z) = 0$ graphically in this 3D ComplexPlot?

My main objective is to show that if $f: \mathbb{R}\to\mathbb{R}$ is defined by $f(x) = x^2 +1$ ($x\in\mathbb{R}$), then the function doesn't intersect with $y = 0$ (or $f(x) = 0$), but it does intersect with $\Re(z) = 0$ if $f: \mathbb{C}\to\mathbb{C}$ is defined by $f(z) = z^2 + 1$ ($z\in\mathbb{C}$).

Answer 1

f[z_] = z^2 + 1;

ComplexPlot3D[f[z], {z, -5 - 5 I, 5 + 5 I},
 AxesLabel -> (Style[#, 12, Bold] & /@ {Re[z], Im[z], Abs[f[z]]}),
 PlotLegends -> Automatic]

f[z] == 0 is not a line but rather two points.

pts = {Re[#], Im[#], 0} & /@ (z /. Solve[f[z] == 0, z])

(* {{0, -1, 0}, {0, 1, 0}} *)

To see this graphically, limit the PlotRange and reverse the z-axis.

Show[
 ComplexPlot3D[f[z], {z, -5 - 5 I, 5 + 5 I},
  AxesLabel -> (Style[#, 12, Bold] & /@ {Re[z], Im[z], Abs[f[z]]}),
  PlotLegends -> Automatic,
  PlotRange -> {0, 1.1},
  ClippingStyle -> None,
  PlotPoints -> 125,
  ScalingFunctions -> {None, None, "Reverse"}],
 Graphics3D[{Black, AbsolutePointSize[4], Point /@ pts}]]

EDIT: To show the line that passes through pts

Show[
 ComplexPlot3D[f[z], {z, -5 - 5 I, 5 + 5 I},
  AxesLabel -> (Style[#, 12, Bold] & /@ {Re[z], Im[z], Abs[f[z]]}),
  PlotLegends -> Automatic,
  PlotRange -> {0, 1.1},
  ClippingStyle -> None,
  PlotPoints -> 125,
  ScalingFunctions -> {None, None, "Reverse"}],
 Graphics3D[{Black, Thick, InfiniteLine[pts]}]]

Answer 2

Using just built-in functions

Show[
  ComplexPlot3D[z^2 + 1, {z, -5 - 5 I, 5 + 5 I}, ColorFunction -> "LightTerrain"], 
  ComplexPlot3D[0, {z, -5 - 5 I, 5 + 5 I}, ColorFunction -> "LightTerrain", PlotStyle -> Opacity[0.5]],
  Graphics3D[{Thick, Red, Line[{{0, -5, 0}, {0, 5, 0}}]}],
  Axes -> True, AxesOrigin -> {0, 0, 0}, 
  AxesLabel -> {Re[z], Im[z], w}, Boxed -> False, 
  PlotRange -> {-10, 50}, BoxRatios -> {1, 1, 0.5}]

Probably a different viewpoint would show better what's going on.

Using David Park's Presentations add-on

This add-on allows, among many other features, working directly with complex graphics objects and complex functions; it includes functions such as ComplexCartesianSurface that predate the functions such as ComplexPlot3D introduced in Mathematica 12.0. (To obtain Park's add-on, see: Where is David Park's Mathematica site?.)

Since Presentations does not include a function that maps points in the complex plane to points in space, let's define a simple version of one:

complexTo3D[pts_, z_] := Append[z] /@ ReIm[pts]

The function:

f[z_] := z^2 + 1

Load Park's add-on:

<< Presentations`

And create the 3D graphic:

With[{zmin = -1 - 1.75 I, zmax = 1 + 1.75 I},
  zeros = z /. Solve[f[z] == 0, z];
  Draw3DItems[
    {
     Opacity[0.85],
     ComplexCartesianSurface[f[z], Abs, {z, zmin, zmax},
       PlotPoints -> 50,
       Mesh -> 12,
       MeshFunctions ->
         {Function[{x, y, z, u, v}, Re[f[u + I v]]],
          Function[{x, y, z, u, v}, Im[f[u + I v]]]},
       MeshStyle -> {White, LightGray},
       ColorFunction -> 
         Function[{x, y, z, u, v}, ColorData["RedGreenSplit"][Rescale[Arg[f[u + I v]], {-\[Pi], \[Pi]}]]],
       ColorFunctionScaling -> False,
     ScalingFunctions -> {None, None, "Reverse"}
     ],
     Opacity[0.3],
     ComplexCartesianSurface[0, Abs, {z, zmin, zmax},
       Mesh -> None, PlotStyle -> LighterGray,
       ScalingFunctions -> {None, None, "Reverse"},
       PlotTheme -> "Classic"
     ],
     PointSize[Large], Red, Point /@ complexTo3D[zeros, 0]
   },
   BaseStyle -> {12, Bold},
   PlotRange -> {-4, 0.75},
   Boxed -> False, Axes -> True, AxesOrigin -> {0, 0, 0}, 
   AxesLabel -> {Re[z], Im[z], Abs[f[z]]},
   Ticks -> False,
   BoxRatios -> {1, 1, 0.75}
  ]
 ]

The mesh on the graph of $\lvert f(z)\rvert$ is formed by level curves of $\Re(f(z)$ and $\Im(f(z)$, while the coloring is obtained from $\arg(f(z))$. (This may be visual overload!)

One could doubtless experiment with the various colors to obtain a more informative, or at least more pleasing, image. Note that I've exploited the enhancement from @BobHanlon's answer of reversing the direction of the f(z)-axis.