Characteristic function using conjugate property

Question

To prove that $e^{−i|x|}$ is not a characteristic function:

$$e^{−i|x|} =\cos|x|-i \sin|x|.$$

Its conjugate will be $\cos|x|+i \sin|x|$ which is not equal to $\phi(-x)$.

Is my solution correct?

Answer 1

Your solution is correct. Maybe you can work with the exponential form since there is no need to switch to algebraic form: let $\phi\colon x\mapsto e^{-i|x|}$; we have $$\overline{\phi(x)}=e^{i|x|}$$ and $$\phi(-x)=e^{-i|-x|}=e^{-i|x|}$$ which is different from $\overline{\phi(x)}$, for example if $x=\pi/2$.