Is expectation of random variable independent of its characteristic function?

Question

For any random variable, does that equation hold? I proved for normal distribution, but I can't generalize.

E$[xe^{itx}] = E[x]E[e^{itx}]$

Thanks in advance.

Answer 1

No, in general the equality does not hold. The left-hand side equals (up to constants) the derivative of the characteristic function. (If the equality was true, then every $X$ with $\mathbb{E}X=0$ would have a characteristic function which is constant.)

Answer 2

Counterexample: let $X$ be a random variable with $P(X=1) = P(X=-1) = 1/2$ (the so-called Rademacher distribution). The left side is $i \sin(t)$ but the right side is 0.