BTW: The method with GetCoordinates mentioned above by Hans is perfect for picking the approximations from the section-plot:

• Right click into the section plot and select GetCoordinates
• For all cross sections of a curve with va0 with a horizontal line h0:
  • Click on the cross section point
  • Ctrl+C
• Create a new empty input cell
  • Ctrl+V
  • Add "{va0, h0}->" in front of the list
  • Enclose the right hand side list rhs into Round[ Flatten[ rhs[[All,1]] ], 10] (or Round[...,1])
• Execute the cell. The output is exactly what you need as an element of solApproxs. You can copy it as Plain Text and paste it into the definition of solApproxs.

By the way: The function becomes more vivid if you plot it in 3D:

(* Show real and imaginary part of function-values in 3D *)
Print[Plot3D[{Re[f[Va, En]], Im[f[Va, En]]}, {Va, 0, 50000}, {En, 
    30000, 70000},
   PlotRange -> {All, All, Automatic},
   PlotPoints -> {50, 40}, MaxRecursion -> 2,
   AxesLabel -> {"Va", "En"}, PlotLegends -> {"Re(f)", Im["f"]}]];

No, you did not define them. There they are variables to be used by ContourPlot, and as far as I know they are local to the plot-routine. So in your Point statement both have no value. Look at your error message, there appears En , without a value of course.

Try your ContourPlot without Point[{Va, En}] and there will be no error.

Yes, you are right, Hans. They just say "The transcendental complex equation (28) has been solved with the use of MATHEMATICA". But they don't publish the code they use to get the results.

I have also tried to contact the author to ask about this through the email address listed in the journal, but to no avail, there has not been an email reply from the author.

And, by the way, @Werner Geiger are you going to make a post about solving this problem like you said a few days ago?

I also feel confused about whether this problem can be solved or not :( However, the results have been shown in the journal. And, Werner also said that he had work on this problem and collected everything in a notebook. Then he will soon come up with a post that includes this notebook.

So, according to Hans, the expression T11 in the paper is actually T22?

However, I have tried to multiply the matrix manually (by hand) to find T11 and the result is the same as T11 in the paper.

Very interesting post and notebook DiracKasten1.nb. Of course you are right concerning the problem posing. We search for solutions of the complex equations of T11 == 0, not Re[T11] == 0 as Affaa told us originally.

I did not check whether T11 should be T22. I believe you are right. But its just naming in our context.

Hence you did not ask for marking points, but for finding them, i.e. solving your problem. See my post dated 01.03.2021, "Point 4: Your actual problem". There I gave you complete answer and coding.

BTW: Meanwhile I read your original paper and found that:

1. Your function definition seems to be wrong.
2. Your problem posing is definetly wrong.

I have resumed work on your problem and collected everything in a notebook. I will soon come up with a post that includes this notebook. It will solve your problem completely and correctly.

OMG! Of course my mini-example has nothing to do with your problem. You asked how to mark and label points in some plot, and I told you.

If you have the coordinates ps = {{va, En}, ...} of the points and their labels lbls you can can mark them with "Point" and label them with "Inset" within the plot' s Epilog . As an example :

Block[{f1, f2, x, y, cnt, ps, lbls},
  (* Define two functions(x) *)
  f1[x_] := x^2;
  f2[x_] := 1/2 + x/4;
  (* Calculate the cross section points *)
  sol = Solve[y == f1[x] == f2[x], {x, y}, Reals];
  Echo[sol];
  ps = {x, y} /. sol;
  cnt = Length[ps];
  (* Build labels for those points *)
  lbls = Row[{"P", #}] & /@ Range[cnt];
  Print["Solution points and labels {ps,lbls} = ", 
   Column[{N[ps], lbls}]];
  (* Plot it *)
  Print[
   Plot[{f1[x], f2[x]}, {x, -1, 1},
    Epilog -> {
      (* The points *)
      PointSize[Large], Point[ps],
      (* The labels *)
      Table[
       Inset[Style[lbls[[i]], Bold, Larger], ps[[i]], {2, 2}], {i, 
        cnt}]
      }
    ]
   ];
  ];

How do I get an expression with the integer value n in the solution or plot? Will it be the same or close to the plot in the source of my journal?

I'm really sorry that my question got confused.

This is the text / resource I am currently studying and I am having trouble with it. The equation I mean is in Chapter II, Section B, more precisely equation (24) and its components are in equations (12) to (17). While the results to be obtained are in chapter III, in plots and tables.

https://arxiv.org/pdf/1309.0308.pdf

@Affaa Nam, posting a question with unclear details and/or purpose can make others confused and will result in unwanted responses. Please take time to think through the answers and suggestions offered by other members here. If you are still in need of an assistance, then make sure you pinpoint your problem and state it as clearly as possible so that others will have a decent chance to assist.

Yes Hans is correct. You do not have an equation. You have an expression (which is only one side of an equation). You are defining a variable f[En,Va] with that expression. You need to show the original problem from the textbook or source that you have because you are missing elements in your questions that will allow us to give you a complete solution. The problem itself does not seem overly difficult, but there does seem to be missing information.

I apologize if my explanation is not understandable, Sir.

It's shorter like this:

I have a transcendental complex equation:

T := (E^(I*q*b)/(16*A^2*B*
       H) (((A + H)^2) ((A + B)^2*
          E^(2*I (k (a - b) - p*a)) - (A - B)^2*
          E^(-2*I (k (a - b) - p*a))) + ((A - H)^2) ((A - B)^2*
          E^(-2*I (k (a - b) + p*a)) - (A + B)^2*
          E^(2*I (k (a - b) + p*a))) + 
      2 (A^2 - B^2) (A^2 - H^2) (E^(2*I*p*a) - E^(-2*I*p*a))));

or in the form of a picture is as follows

I consider the T function to be f[En_] because the value I want to find is the value of En.

The known variables are:

c = 137.036;
Vb = 50000;
a = 0.01;
b = 0.02;
k = Sqrt[En^2 - c^4]/c; 
p = Sqrt[(En + c^2 - Va) (En - c^2 - Va)]/c; 
q = Sqrt[(En + c^2 - Vb) (En - c^2 - Vb)]/c; 
A = Sqrt[(En - c^2)/(En + c^2)]; 
B = Sqrt[(En - c^2 - Vb)/(En + c^2 - Vb)]; 
H = Sqrt[(En - c^2 - Va)/(En + c^2 - Va)];

The transcendental complex equation is solved using Wolfram Mathematica and the solution is shown in the form of a plot and table that looks like the following.

and

What I want is to know the coding used to get the plot and table results as I show them. How?

Thank you very much also Mr. Werner Geiger. I will try to understand the description that you give. This really adds to my knowledge, who is new to using Wolfram Mathematica.

Well, as Hans asked in his reply to this question, about what if I tell you what my problem really is, I'll tell you.

So, I have a transcendental complex equation and its variables which I mention as my initial coding. Then I coded it to bring up the plot and found the result I was looking for. The reference plot that I want to make is like the following image,

However, the plot that I produce from my coding is still very much different from the shape of the image I want. Here's a plot of the results of my coding

I have the assumption that the plot line that matches my reference plot is the blue plot line, that is, the plot of the "Real" section. Then I tried again to make a new plot by removing the coding leading to the "Imaginary" section so that the resulting plot was "Real" only. However, an error appears in the output. After I saw the coding given by Werner Geiger, it turned out that it was quite complicated to use the coding used to bring up the "Real" only, not as simple as my assumption which was only by deleting the codes that referred to the "Imaginary" section.

Furthermore, in my reference plot, there are points along the lines which I will write in tabular form as follows.

The values ​​in the table are En values. That's why I initially asked about how to find the value of En at the point where Va intersects with En, or the value of En on the blue line ("Real") on the plot.

Related to simplifying a function manually, I have tried to simplify or convert it in trigonometric form so that it can be separated between the Real part and the Imaginary part. However, the form of the equation is still complicated, in my opinion. So that the equation that I use in my coding is still the original equation, not the one in trigonometric form.

Like this is actually my problem. I'm having a hard time plotting the results for solving the equation and writing the values ​​down in a table.

As Werner pointed out your f[En_] is somewhat strange.

Type for example (after your code has run) f[50000]. What do you get?

What about telling us what exactly your problem is. Perhaps there is an alternative way to find a solution?

Can the coordinate values be written together in a table, Sir?

Go with the pointer unto your plot, then do a right mouse-click, a window opens. click Get Cooridinates and you can chose any region of the plot with your mouse.

Va is in the range 0 to 50000.

En is the value I want to find and is in the range 30000 to 70000.

I define Va and En in the form {Va, 0, 50000}, {En, 30000, 70000}, located in line with the code "ContourPlot {bla bla bla"