Simplifing equation with complex coefficients

Question

I'm trying to simplify an expression with complex coefficients. It should be very straightforward, but for some reason, I cannot get the correct answer. Please, see the code below:

Simplify[\[Gamma]0[0] == Re[(0.5852908990577314` + 1.831942535298071` I)*\[Gamma]0[
      1] + (0.25530515981369933` + 0.6158107565352037` I)*\[Gamma]0[
      3]], {Element[\[Gamma]0[1], Reals], 
  Element[\[Gamma]0[3], Reals]}]

It should return

0.585291\[Gamma]0[1] + 0.255305\[Gamma]0[3] == \[Gamma]0[0]

But returns

Re[(0.585291 + 1.83194 I) \[Gamma]0[
     1] + (0.255305 + 0.615811 I) \[Gamma]0[3]] == \[Gamma]0[0]

Can't really find my error. Thank you for your help!

Answer 1

Use ComplexExpand

$Version

(* "14.1.0 for Mac OS X ARM (64-bit) (July 16, 2024)" *)

Clear["Global`*"]

γ0[0] == 
  Re[(0.5852908990577314` + 1.831942535298071` I)*γ0[1] + 
      (0.25530515981369933` + 0.6158107565352037` I)*γ0[3]] // 
    ComplexExpand

(* γ0[0] == 0. + 0.585291 γ0[1] + 0.255305 γ0[3] *)

% // Rationalize

(* γ0[0] == 0.585291 γ0[1] + 0.255305 γ0[3] *)

Answer 2

eqn = γ0[0] == (0.5852908990577314` + 1.831942535298071`  I)*γ0[1] + 
               (0.25530515981369933` + 0.6158107565352037`  I)*γ0[3];

Using ReplaceAll:

Reverse[eqn /. x_Complex :> Re[x]]

0.585291 γ0[1] + 0.255305 γ0[3] == γ0[0]