# Is there a way to simplify a multivalued function?

A minimal example is

FullSimplify[Exp[-I*x] Sqrt[Exp[I*2 x]], Assumptions -> {x > 0}]


which will just return the same input. However, what I expect to see is \pm 1 (here \pm means plus or minus)because Sqrt[Exp[I*2 x]] can be factored out to be Exp[I*(x+n* Pi)] and further equals (\pm 1) Exp[I*x].

If that is not possible, can I write some my own simplification rules to simplify this multivalued function, which only takes one of the roots, for example Exp[I*x]?

• Exp[-I*x] Sqrt[Exp[I*2 x]] // PowerExpand[#, Assumptions -> x > 0] &  gives E^(I Pi Floor[1/2 - x/Pi])  and E^(I Pi Floor[1/2 - x/Pi]) // FullSimplify[#, Assumptions -> {x > 0}] &  gives (-1)^Floor[1/2 - x/Pi]  – Akku14 May 23 '18 at 18:07
• Thank you! Would you please write it as an answer so that I can pick you as the correct answer? – Jake Pan May 23 '18 at 18:35

Exp[-I*x] Sqrt[Exp[I*2 x]] // PowerExpand[#, Assumptions -> x > 0] &