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2

Coefficient[E^(I a (t - b)) // ExpandAll, E^(I a t)] (* Exp[-I a b] *)


3

The following gives what you intended: Refine[Expand[P[x, y]^2], (x|y|beta) \[Element] Reals] (* ==> Conjugate[z[y]]^2/E^((2*I)*beta*x) + 2*Conjugate[z[y]]*z[y] + E^((2*I)*beta*x)*z[y]^2 *) In cases where you can live with expansion of complex exponentials into Sin and Cos you can also use ComplexExpand[P[x, y]^2, z[y], ...



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