The following question comes from M. Kardar's "Statistical Physics of Particles". Let $x$ be a random variable with pdf $p(x)$. Kardar defines the "characteristic function" as the Fourier transform of $p(x)$; i.e.
$$ \tilde p(k)=\int dx\,p(x)\,e^{-ikx} $$
I see why this is a useful definition: by differentiating both sides wrt $k$, we generate the moments of $p(x)$ (after setting $k=0$). Without further motivation, Kardar then introduces the "cumulant generating function"
$$ \ln\tilde p(k)=\sum_{n=1}^\infty \frac{(-ik)^n}{n!}\langle x^n\rangle_c $$
where $\langle\cdot\rangle_c$ denotes cumulant. Now, I understand we can exponentiate the above and equate the two expressions for $\tilde p(k)$, and thereby relate moments to cumulants. My question is: why is the cumulant generating function defined this way? Is it just because it yields such familiar results as
$$ \langle x^2\rangle_c =\langle x^2\rangle - \langle x\rangle^2 $$
Is there any more intuition behind this defintion?