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As with the real exponential function (see {{slink||Functional equation}} above), the complex exponential satisfies the functional equation
<math display=block>\exp(x+y)= \exp(x)\cdot \exp(y).</math>
Among complex functions, it 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 |year=1959 |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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