= Solution
A <characteristic function> is useful for proving convergence in distribution, including for laws without a finite <moment-generating function>: the continuity theorem says that pointwise convergence of characteristic functions to a <characteristic function> continuous at zero implies convergence in distribution to its law. For example, the <Cauchy distribution> has a <characteristic function> although its <moment-generating function> is not finite on any neighbourhood of zero.
A <moment-generating function> is especially useful for exponential tail bounds: for $t>0$ where it is finite, <Markov's inequality> gives
$$
\boxed{\Pr(X\geq a)\leq e^{-ta}M_X(t).}
$$
Optimizing over $t$ yields a <Chernoff bound>. The positive real exponential in this argument permits the inequality; a unit-modulus complex exponential does not give the corresponding tail bound.
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