Choose a branch cut using assumptions

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? 1 Answer 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.)

• Yes... Thank you for pointing that out. Also just a quick thing: can I ask if there is a difference in using Simplify[] versus // Simplify? – Histoscienology Jan 9 at 4:09
• @Histoscienology No, Simplify[x], Simplify@x (prefix), and x//Simplify (postfix) are all equivalent (although they have different precedence rules). See here for more information. – Doorknob Jan 9 at 4:15