# Simplifying expressions involving Sqrt[] inside ComplexConjugate[]

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

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


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 + 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[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[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[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[Conjugate][
x Sqrt[-4 x^2 + z^2]])}