How can we find the appropriate values of unknowns such that the following expression must be a real number? FindInstance

Question: I have some mathematical functions including various parameters. I want to find instances for the parameters such that the function must be in a real domain after substituting the values of parameters into the mathematical functions.

For example;

expr = 3^(1/4)*Sqrt[(I*a*b*Sqrt[d]*Sqrt[v]*Sech[(1/2)*Sqrt[a]*((-t)*v + x)]^2)/(Sqrt[a*c*\[Alpha] - 8*c*\[Kappa] + 4*c*\[Alpha]*\[Kappa]^2]*(b^2 - a*d*(1 + Tanh[(1/2)*Sqrt[a]*((-t)*v + x)])^2))]


How can we find the appropriate positive integer values of unknowns v, \[Alpha], \[Beta], \[Kappa], \[CapitalPhi] such that the following expression must be a real number? (where the independent variables x and t are real numbers.)

An example for Manual selection of the parameters:

If we select

parameters={v -> 1, \[Alpha] -> 1, \[Beta] -> 2, \[Kappa] -> 1, \[CapitalPhi] -> 2, c -> 1, a -> 2, b -> 3, d -> 1},

and expr /. parameters, we get

2^(1/4)*3^(3/4)*Sqrt[Sech[(-t + x)/Sqrt[2]]^2/(9 - 2*(1 + Tanh[(-t + x)/Sqrt[2]])^2)]

How can we achieve this in Mathematica 13.2?

In my opinion,

Step.1: firstly we will find the terms including square roots in expr.

Step.2: And we will create conditions such that the inside of the roots should be equal or greater than zero.

Step.3: And lastly, we will use FindInstance such that the conditions hold.

Try FunctionDomain:

FunctionDomain[ {
Sqrt[-((I*Sqrt[v]*Sech[((-t)*v + x)*Sqrt[a]]*a*Sqrt[c])/(Sqrt[-8*\[Kappa]*c +4*\[Alpha]*\[Kappa]^2*c + \[Alpha]*c*a]*((-Sech[((-t)*v + x)*Sqrt[a]])*b + Sqrt[c^2 - 4*a*b])))],
Element[{a, b, c, v, \[Alpha], \[Kappa]}, Integers]},
{a, b, c,v, \[Alpha], \[Kappa]}, Reals ]
(*False*)


Unfortunately result is False, Mathematica cannot find an instance!

• Thank you. I have edited the post.
– RF_1
Commented Apr 12, 2023 at 8:18