Let $X$ be a random variable with density $f$ and characteristic function $\varphi$. Say we know $\varphi$ up to a constant $c$. Is it possible to evaluate this constant using $\int f(x)dx=1$ (or by another method)?
(Motivation for question: I was thinking about this question because of the following type of question: Say $X$ is a random variable with (uniform) density $f(x)=c$ with support $[0,5]$. Calculate $c$.)
If $\phi(t):=\mathbb Ee^{itX}$ then $\phi(0)=1$.
So if $\chi(t)$ denotes the characteristic function up to a constant then: $$\phi(t)=\frac{\chi(t)}{\chi(0)}$$