0
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 ClearAll["Global`*"]
zz = FullSimplify[-(((0. + 396232. I) x Cos[
         0.000198116 Sqrt[x^2]] Csc[(Sqrt[157/10] Sqrt[x^2])/20000])/
      Sqrt[x^2]) - (I (x^2)^(3/4))/(4945.28 x Csc[
        0.186346 (x^2)^(1/4)] Sin[0.0621153 (x^2)^(1/4)] Sin[
        0.124231 (x^2)^(1/4)] - 
      4945.28 x Csch[0.186346 (x^2)^(1/4)] Sinh[
        0.0621153 (x^2)^(1/4)] Sinh[0.124231 (x^2)^(1/4)])]
Plot[{Re[zz], Im[zz]}, {x, 0, 100000}]
FindRoot[zz == 0, {x, 8000}]

I wanted to find the root of this equation but seems like NSolve is not working for these kinds of equation. How to deal about these kind of situation. I tried NSolve but it is failing. Findroot function is working but demands initial guess all the time. How to implement NSolve for this kind of problem.

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2 Answers 2

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You can specify the region where to look for the solutions:

NSolve[0 == zz && Abs[x] < 100000, x]

(*{{x -> -87215.4 + 0. I}, {x -> -71358. + 0. I}, {x -> -55500.7 + 
0. I}, {x -> -39643.3 + 0. I}, {x -> -23786. + 
0. I}, {x -> -7928.67 + 0. I}, {x -> 7928.67 + 0. I}, {x -> 
23786. + 0. I}, {x -> 39643.3 + 0. I}, {x -> 55500.7 + 0. I}, {x ->
71358. + 0. I}, {x -> 87215.4 + 0. I}}*)
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    $\begingroup$ Tricky solution! The result is real and I'm wondering why NSolve[0 == zz && 0 < x < 100000, x , Reals] fails. $\endgroup$
    – Ulrich Neumann
    Commented Nov 17, 2018 at 13:11
  • 1
    $\begingroup$ According to the manual, In NSolve[expr,vars,Reals] all variables, parameters, constants, and function values are restricted to be real. So technically, since the function is complex valued, you should use NSolve[0 == z && x \[Element] Reals && 0 < x < 100000, x, Complexes]. Nevertheless it fails. I would guess it has something to do with the way MMA deals with expressions like $\sqrt{x^2}$. $\endgroup$
    – yuriyi
    Commented Nov 18, 2018 at 14:26
  • $\begingroup$ @ yuriyi Thanks! $\endgroup$
    – Ulrich Neumann
    Commented Nov 18, 2018 at 14:45
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I don't know why NSolve fails. Here is a workaround which uses the datapoints used in Plot

pict = Plot[{Im[zz], 0}, {x, 0, 100000}, MaxRecursion -> 5 ] 
(* added "0" as second argument *)

np = Graphics`Mesh`FindIntersections[pict[[All, 1]]  ];
(* intersection points Im[zz] and 0 *)
(*{{7929.56, 7.27596*10^-12}, {23786.7, 0.}, {39643.5,7.27596*10^-12}, 
{55500.7, -3.63798*10^-12}, {71358., 0.}, {87215.5,0.}}*)

These are the points Im[zz]==0!

Show[{pict, Graphics[{Red, PointSize[Large], Point[np]}]}]

enter image description here

If you need more precision you can take the result as Starting values inside FindRoot.

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