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→Complex exponential: Removing apparently false statement. For example, b^{w+z} = b^w b^z and b^0 = 1 for any b > 0, not just b = e. |
m →Complex exponential: use z instead of x |
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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
<math display=
Among complex functions, it is the unique solution which is [[holomorphic]] at the point {{tmath|1= z = 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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