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Cauchy's [[exponential functional equation]] is
<math display="block">f(w+z)=f(w)f(z).</math>
Among continuous real functions, its only solutions are {{tmath|1=\textstyle f(x) = e^{ax} }}. Among complex functions, the complex exponential is the unique solution which is [[holomorphic]] at the point {{tmath|1= x = 0}} and takes the derivative {{tmath|1}} there.<ref>{{cite book |last=Hille |first=Einar |title=Analytic Function Theory |volume=1 |place=Waltham, MA |publisher=Blaisdell |chapter=The exponential function |at=§ 6.1, {{pgs|138–143}} }}</ref>
The [[complex logarithm]] is a [[left inverse function|right-inverse function ]] of the complex exponential:
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