Exponential function: Difference between revisions

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===Functional equation===

''The exponential satisfies the [[functional equation]]:''

<math display=block>\exp(x+y)= \exp(x)\cdot \exp(y).</math>

This results from the uniqueness and the fact that the function

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<math display="block">e^z = \lim_{n\to\infty}\left(1+\frac{z}{n}\right)^n</math>

As with the real exponential function (see {{slink||Functional equation}} above), the complex exponential satisfies the functional equation

Cauchy's [[exponential functional equation]] is

<math display="block">f\exp(w+z+w)=f \exp(wz)f\cdot \exp(zw).</math>

Among continuous real functions, its only solutions are {{tmath|1=\textstyle f(x) = e^{ax} }}. Among complex functions, the complex exponentialit is the unique solution which is [[holomorphic]] at the point {{tmath|1= xz = 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: