\overline{ e^{ik(x)}} = e^{-ik(x)}

this definitely works for k as a constant but i'm not sure how to prove with k(x)

works for k \in \mathbb{R}

@jeffcliff The only chance for e^[ik(x)] to equal e^[−ik(x)] is to select x for which k(x) equals either 0 or π.

@chuculate note the bar should be rendering above representing complex conjugacy. it should be valid for all real k(x)

@jeffcliff maybe from their trigonometric expressions
since
conj(e^[ik(x)]) = cos (k(x)) + i · sin (k(x))
and
e^[-ik(x)] = cos (-k(x)) - i · sin(-k(x))
Since cosine is an even function, its argument sign will not matter.
Since sine is an odd function, sin(-x) is equivalent to -sin(x) and thus that part also becomes the same, above and below?

@chuculate @jeffcliff
\overline{e^{iz}} = \overline{e^{i(a+ib)}} = \overline{e^{i(a+ib)}} = \overline{e^{ia-b}}=e^{-b}\overline{e^{ia}}=e^{-b}e^{-ia} = e^{i(a-ib)} Its not true in general