Simplify complex square roots

Question

I want to simplify the following expression as follows: $$\sqrt{\left(\gamma -\kappa _2\right){}^2-16 G_2^2}=i\sqrt{-\left(\gamma -\kappa _2\right){}^2+16 G_2^2}$$

I am trying

FullSimplify[Sqrt[(γ - Subscript[κ, 2])^2 - 16*Subscript[G, 2]^2],
  {Subscript[κ, 2] > 0, Subscript[G, 2] > 0, γ > 0, (γ - Subscript[κ, 2])^2 < 16*Subscript[G, 2]^2}
 ]

but it returns exactly the same thing. This is part of a more complicated expression.

Answer 1

As Virgil stated, what you want here is Refine.

Refine[
 Sqrt[(γ - κ)^2 - 16*G^2], {κ > 0, 
  G > 0, γ > 0, (γ - κ)^2 < 16*G^2}]
(* I Sqrt[16 G^2 - (γ - κ)^2] *)

What is the difference between FullSimplify and Refine, when they are both given the same Assumptions? From R.M.'s answer, we learn

The primary difference between Refine and the two *Simplify functions is that Refine only evaluates the expression according to the assumptions given. It might so happen to be the simplest form when evaluated, but it does not check to see if it is indeed the simplest possible form. You should use Refine when your goal is not to simplify the expression but to just see how the assumptions transform it (e.g., square root of a positive quantity).

Simplify, on the other hand, performs basic algebraic simplifications and transformations to arrive at the "simplest" result. Refine is one among them, and is also mentioned in its doc page. Here, "simplest" might not necessarily fit your definition of simple. It is what appears simple to Mathematica, and that is defined by LeafCount.

Answer 2

It depends upon your ultimate aim. If this is the very last expression of yours that you will not transform any more, but will only look at, then you may do something as follows:

 HoldForm[I]*Sqrt[(-1)*((\[Gamma] - Subscript[\[Kappa], 2])^2 - 
     16*Subscript[G, 2]^2)]

If it is an intermediate step, and you will transform the expression further, I recommend to do nothing, since Mma transforms everything according its own understanding of the simplicity, and the efforts to make it looking differently will be much too high.