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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]?

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  • $\begingroup$ 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] $\endgroup$
    – Akku14
    May 23, 2018 at 18:07
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    $\begingroup$ Thank you! Would you please write it as an answer so that I can pick you as the correct answer? $\endgroup$
    – Jake Pan
    May 23, 2018 at 18:35

1 Answer 1

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With square roots, PowerExpand with conditions often helps.

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

(*   E^(I Pi Floor[1/2 - x/Pi])    *)

E^(I Pi Floor[1/2 - x/Pi]) // FullSimplify[#, Assumptions -> {x > 0}] & 

(*   (-1)^Floor[1/2 - x/Pi]   *)
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