Question: I am working on a physics reserach project for school and I have run into some troubles working Mathematica. I am a fairly inexperienced mathematica user so any help would very much be appreciated.

I need to find the roots of the transcendental equation 
$$\zeta\_n \tan(\zeta) - \sqrt{R^2-\zeta^2\_n}=0$$
and then collect them into a list, $\{\zeta\_n\}$, which I can sum over.

FindRoot works but only finds one root at a time. For example $R^2=18$ gives

> 
> FindRoot[Sqrt[18. - zeta^2] - zeta Tan[zeta] == 0, {zeta, 3}]
> 
> 
> {zeta -> 3.66808}
>  From a plot of the function I know there should be two roots for this particular value of $R$, however FindRoot only gives one.
> 
> 
> 

I have had no luck with NSolve or Solve either. 

> 
> NSolve[xi[zeta, Erg4]- zeta\*Tan[zeta] == 0, zeta]
> 
> 
> NSolve[Sqrt[18. - zeta^2] - zeta Tan[zeta] == 0, zeta]
> 
> 
> 

Also once I have succeeded in finding all roots how does one put them in an indexed set ${\zeta\_n}$.

Answer: 
```
With[{maxR = 10},
 Manipulate[
  expr = Sqrt[R^2 - zeta^2] - zeta Tan[zeta];
  zero = zeta /. NSolve[{expr == 0,
       R^2 - zeta^2 >= 0}, zeta] // Quiet;
  Column[{
    Plot[expr, {zeta, -maxR, maxR},
     Exclusions -> {Cos[zeta] == 0},
     PlotRange -> {{-maxR, maxR}, {-25, 40}},
     Epilog -> {Red, AbsolutePointSize[4],
       Point[{#, 0} & /@ zero]},
     AspectRatio -> 1],
    zero},
   Alignment -> Center],
  {{R, Sqrt[18]}, 0, maxR, Appearance -> "Labeled"}]]

```

![enter image description here](https://i.stack.imgur.com/8xcBF.png)
If you only want real solutions, then `NSolve` will find them if we help it by including the domain for `zeta`.

```
R = Sqrt[18];
sol = NSolve[Sqrt[R^2 - zeta^2] - zeta Tan[zeta] == 0 && -R <= zeta <= R]
(*
  {{zeta -> -3.66808}, {zeta -> -1.26743},
   {zeta -> 1.26743},  {zeta -> 3.66808}}
*)

```

You can save them in an indexed list like this:

```
roots = zeta /. sol
(*  {-3.66808, -1.26743, 1.26743, 3.66808}  *)

```