Simplifying expressions involving Sqrt[] inside ComplexConjugate[]

Question

I have a matrix mat with expressions involving square-root with complex-conjugation operation which makes the simplification difficult.

mat = {{Conjugate[
 4 (1 + E^(y Sqrt[-4 x^2 + z^2])) x^2 + 
  z (-(1 + E^(y Sqrt[-4 x^2 + z^2])) z + (-1 + E^(
        y Sqrt[-4 x^2 + z^2])) Sqrt[-4 x^2 + z^2])], 
E^(-(1/2)
   y (2 z + Sqrt[-4 x^2 + z^2] + 
   Conjugate[Sqrt[-4 x^2 + z^2]]))}, {E^(-(1/2)
   y (2 z + Sqrt[-4 x^2 + z^2] + Conjugate[Sqrt[-4 x^2 + z^2]])), 
Conjugate[x Sqrt[-4 x^2 + z^2]] + 
 2 I (-1 + E^(1/2 y Sqrt[-4 x^2 + z^2])) }};

How can we instruct Mathematica to do the simplification with following conditions:

Step 1. Replace $x$ by $x+\epsilon$ such that $\epsilon$ is very small and neglect $\epsilon^2$ terms

Step 2. After Step 1, put $z^2 = 4 x^2$ and simplify the result

Answer 1

We may do this using a Taylor series. First we write the matrix as a function of eps:

mat = {{Conjugate[
     4 (1 + E^(y Sqrt[-4 x^2 + z^2])) x^2 + 
      z (-(1 + E^(y Sqrt[-4 x^2 + z^2])) z + (-1 + 
            E^(y Sqrt[-4 x^2 + z^2])) Sqrt[-4 x^2 + z^2])], 
    E^(-(1/2) y (2 z + Sqrt[-4 x^2 + z^2] + 
         Conjugate[Sqrt[-4 x^2 + z^2]]))}, {E^(-(1/2) y (2 z + 
         Sqrt[-4 x^2 + z^2] + Conjugate[Sqrt[-4 x^2 + z^2]])), 
    Conjugate[x Sqrt[-4 x^2 + z^2]] + 
     2 I (-1 + E^(1/2 y Sqrt[-4 x^2 + z^2]))}};
f[eps_] = mat /. x -> x + eps;

Then we expand this into a Taylor series (more accurate: Maclaurin series)

f[0] + eps D[f[x1], x1] /. x1 -> 0 // FullSimplify

{{Conjugate[
    4 (1 + E^(y Sqrt[-4 x^2 + z^2])) x^2 + 
     z (-((1 + E^(y Sqrt[-4 x^2 + z^2])) z) + (-1 + E^(
           y Sqrt[-4 x^2 + z^2])) Sqrt[-4 x^2 + z^2])] - (1/
   Sqrt[-4 x^2 + z^2])
   4 eps x (-z - 2 Sqrt[-4 x^2 + z^2] + 
      E^(y Sqrt[-4 x^2 + z^2]) (4 x^2 y + z - 2 Sqrt[-4 x^2 + z^2] + 
         y z (-z + Sqrt[-4 x^2 + z^2]))) Derivative[1][Conjugate][
     4 (1 + E^(y Sqrt[-4 x^2 + z^2])) x^2 + 
      z (-((1 + E^(y Sqrt[-4 x^2 + z^2])) z) + (-1 + E^(
            y Sqrt[-4 x^2 + z^2])) Sqrt[-4 x^2 + z^2])], (1/
  Sqrt[-4 x^2 + z^2])
  E^(-y (z + Re[Sqrt[-4 x^2 + z^2]])) (Sqrt[-4 x^2 + z^2] + 
     2 eps x y (1 + 
        Derivative[1][Conjugate][Sqrt[-4 x^2 + z^2]]))}, {(1/
  Sqrt[-4 x^2 + z^2])
  E^(-y (z + Re[Sqrt[-4 x^2 + z^2]])) (Sqrt[-4 x^2 + z^2] + 
     2 eps x y (1 + Derivative[1][Conjugate][Sqrt[-4 x^2 + z^2]])), 
  1/Sqrt[-4 x^2 + 
    z^2] (2 I (-Sqrt[-4 x^2 + z^2] + 
        E^(1/2 y Sqrt[-4 x^2 + z^2]) (-2 eps x y + 
           Sqrt[-4 x^2 + z^2])) + Abs[-4 x^2 + z^2] Conjugate[x] + 
     eps (-8 x^2 + z^2) Derivative[1][Conjugate][
       x Sqrt[-4 x^2 + z^2]])}

Answer 2

mat = {{Conjugate[
     4 (1 + E^(y Sqrt[-4 x^2 + z^2])) x^2 + 
      z (-(1 + E^(y Sqrt[-4 x^2 + z^2])) z + (-1 + 
            E^(y Sqrt[-4 x^2 + z^2])) Sqrt[-4 x^2 + z^2])], 
    E^(-(1/2) y (2 z + Sqrt[-4 x^2 + z^2] + 
         Conjugate[Sqrt[-4 x^2 + z^2]]))}, {E^(-(1/2) y (2 z + 
         Sqrt[-4 x^2 + z^2] + Conjugate[Sqrt[-4 x^2 + z^2]])), 
    Conjugate[x Sqrt[-4 x^2 + z^2]] + 
     2 I (-1 + E^(1/2 y Sqrt[-4 x^2 + z^2]))}};

step1 = ExpandAll[ComplexExpand[mat] /. x -> (x + ϵ)] /. Power[ϵ, t_] /; t >= 2 :> 0;
step2 = Simplify[step1 /. z^2 :> 4 x^2];

If you remove the ComplexExpand and use FullSimplify you get this:

{{4 Conjugate[x]^2 - Conjugate[z]^2 + 4 Conjugate[ϵ]^2 + 
   Conjugate[
    8 (1 + E^(2 Sqrt[2] y Sqrt[-x ϵ])) x ϵ + 
     2 Sqrt[2] (-1 + E^(
        2 Sqrt[2] y Sqrt[-x ϵ])) z Sqrt[-x ϵ]], 
  E^(-y (z + 2 Sqrt[2] Re[Sqrt[-x ϵ]]))}, {E^(-y (z + 
     2 Sqrt[2] Re[Sqrt[-x ϵ]])), 
  2 I (-1 + E^(Sqrt[2] y Sqrt[-x ϵ])) + 
   2 Sqrt[2] Conjugate[Sqrt[-x ϵ] (x + ϵ)]}}

... and near $\epsilon=0$ this looks like:

step2 /. ϵ -> 0
(**
  {{4 Conjugate[x]^2 - Conjugate[z]^2, E^(-y z)}, {E^(-y z), 0}}
**)