= Exponential form of Lévy characteristic functions
{title2=$\varphi_{X_t}(u)=e^{t\eta(u)}$}
<Independent increments> and <stationary increments> make a <Lévy process>'s <characteristic functions> multiplicative in time. Together with <continuity of Lévy characteristic functions> and the value one at time zero, this forces an exponential. A continuous local <complex logarithm> is additive: its failure of additivity would be a continuous integer multiple of $2\pi i$, hence zero. The <Cauchy functional equation> and subdivision then give the exponential at every time. This construction of the <characteristic exponent of a Lévy process> precedes the <Lévy–Khintchine formula> describing the exponent's possible form.
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