Solve this equation explicitly for $\beta$:
$$T_{12}=\frac{x_{11}+x_{22}}{1+|\beta|^2}\beta$$
$x_{11}$ and $x_{22}$ are positive real numbers where $x_{11}>x_{22}$ and $T_{12}$ and $\beta$ are complex numbers. and $|\beta|<1$
Up to know I've tried this:
sol1 = Solve[
Subscript[T, 12] == (Subscript[x, 11] + Subscript[x, 22])/(1 + Abs[β]^2) β &&
{Subscript[x, 11], Subscript[x, 22]} ∈ Reals &&
{Subscript[T, 12], β} ∈ Complexes,
β
]
But it gives a complicated solution whereas the answer should be
$$\beta=\frac{T_{12}}{|T_{12}|}\sqrt{\frac{x_{22}}{x_{11}}}$$
according to the scientific article that I'm studying.
sol1 = Block[{$Assumptions = Subscript[x, 11] > Subscript[x, 22] > 0 && {Subscript[x, 11], Subscript[x, 22]} \[Element] Reals && Subscript[T, 12] \[Element] Complexes}, Solve[Subscript[T, 12] == ((Subscript[x, 11] + Subscript[x, 22]) \[Beta])/(1 + Abs[\[Beta]]^2), \[Beta], Complexes]]
seems to work, but addingAbs[\[Beta]] < 1
inSolve
takes too long. $\endgroup$