If a random variable $x$ has a characteristic function $\phi(\omega)$, then the $n^{\mathrm{th}}$ moment of the distribution of $x$, $\mu_n$ can be calculated as:
$\mu_n = \imath^{-n}\left[\frac{d^n}{d\omega^n}\phi(\omega)\right]_{\omega=0}$
Is there a similar formula to compute the $n^{\mathrm{th}}$ cumulant of the distribution, $\kappa_n$ in terms of the characteristic function? Wolfram MathWorld lists a formula as:
$\ln[\phi(\omega)]=\sum_{n=0}^\infty\kappa_n \frac{(\imath \omega)^n}{n!}$
However, this is not helpful in obtaining the $n^\mathrm{tm}$ cumulant (or if it is, I don't know how).