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I am trying to do symbolic manipulations in MMA and I need it to take a complex conjugate of functions with real arguments, like:

Refine[Conjugate[(I^(2 mg))], mg \[Element] Reals]
Refine[Conjugate[BesselJ[mg, x]], {mg \[Element] Reals, x \[Element] Reals}]

I can't find out how to force MMA to actually take the Conjugate...

Please, help!

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A job for ComplexExpand.

Conjugate[(I^(2 mg))] // ComplexExpand
(* Cos[mg \[Pi]] - I Sin[mg \[Pi]] *)
Conjugate[BesselJ[mg, x]] // ComplexExpand
(* -I Im[BesselJ[mg, x]] + Re[BesselJ[mg, x]] *)
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