Why $|e^{itx}| = 1$?

Question

I'm studying some demonstrations of properties of characteristic function in which I have to use that $|e^{itx}| = 1$ but I don't understand it at all. Could you give a clue to demonstrate it?

Answer 1

A key property of the exponential map is

$$e^ze^{z^\prime} = e^{z+z^\prime}$$

This can be proven using Cauchy product.

Based on that, you get for $y \in \mathbb R$

$$\vert e^{iy}\vert^2= e^{iy} \overline{e^{iy}}=e^{iy}e^{-iy}=e^{iy-iy}=1$$

Answer 2

Hint: $$e^{itx}=\cos(tx)+i\sin(tx)$$