Choose a branch cut using assumptions

Question

I understand Sqrt function won't simplify expressions such as $\sqrt{x^2}$ unless there is an assumption that $x \in \mathbb{R}$. In my computation, I get an output containing an expression like this after FullSimplify:

$$e^{i\phi}-\sqrt{e^{2i\phi}}$$

I attempted to use Assumptions as mentioned as above but it still won't simplify. The exact code I used is this:

$Assumptions = 2*π > {ϕ} > 0 && ϕ ∈ Reals ;

Any idea what I did wrong? Or any other solutions?

Answer 1

Your assumptions aren't quite strict enough -- for example, if $\phi=\frac{3\pi}2$, then $e^{i\phi}=-i$, but $\sqrt{e^{2i\phi}}=i$.

(Also, you have an error in your $Assumptions expression; $\phi$ should not have list brackets around it.)

Try instead:

$Assumptions = -π/2 < ϕ ≤ π/2;
E^(I ϕ) - Sqrt[E^(2 I ϕ)] // Simplify
(* => 0 *)

(You don't have to explicitly state that $\phi\in\mathbb R$, because the fact that it can be compared in an inequality already implies that it's real; the field of complex numbers is not ordered.)