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I have
In[57]:= g[a_, x_, y_] =
Simplify[Limit[f4[t], t -> Infinity]] +
Sqrt[-1]*Simplify[Limit[f5[t], t -> Infinity]]
Out[57]= ConditionalExpression[(1 + a + a x + a^2 x - a y)/(
3 + 3 a + a^2) + (I (1 + a x + 2 a y + a^2 y))/(
3 + 3 a + a^2), (x | y) \[Element] Reals && a > 0]
Now I want to express the g[t,z] in terms of a new variable
z = x + Sqrt[-1] y;
I was looking for expression like this
Perhaps that's the answe to your question
expr=(1 + a + a x + a^2 x - a y)/(3 + 3 a +a^2) + (I (1 + a x + 2 a y + a^2 y))/(3 + 3 a + a^2);
expr /. {x -> Re[z], y -> Im[z]} // FullSimplify
(* ((1 + I) + a + a ((2 + I) + a) z - a Re[z])/(3 + a (3 + a))*)
expr /. {x -> Re[z], y -> Im[z]} // FullSimplify (drop the I for y)
$\endgroup$
ComplexExpand[z, z] (*I Im[z] + Re[z]*)
$\endgroup$
Jun 13, 2019 at 8:11
z = x + Sqrt[-1] y. So for {Re[z], Im[z]} // ComplexExpand you get {x, y}. These are the usual definitions of $x$ and $y$ being the real and imaginary parts of $z$, and both $x$ and $y$ being real numbers.
$\endgroup$
FullSimplify you get the OP's desired form.
$\endgroup$
exp = ConditionalExpression[(1 + a + a x + a^2 x - a y)/(3 + 3 a +
a^2) + (I (1 + a x + 2 a y + a^2 y))/(3 + 3 a + a^2), (x | y) ∈ Reals && a > 0];
Simplify[exp, z == x + Sqrt[-1] y]
ConditionalExpression[((1 + I) + a + I a y + a ((1 + I) + a) z)/( 3 + 3 a + a^2), (x | y) ∈ Reals && a > 0]
