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

The functional equation

<math display="block">e^{w+\exp(z}+w)=e^we^ \exp(z)\cdot \exp(w).</math>

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>

holds for every complex numbers {{tmath|w}} and {{tmath|z}}. The complex exponential is the unique [[continuous function]] that satisfies this functional equation and has the value {{tmath|1}} for {{tmath|1=z=0}}.

The [[complex logarithm]] is a [[left inverse function|right-inverse function ]] of the complex exponential:

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<math display="block">\overline{e^z}=e^{\overline z}.</math>

Its modulus is

<math display="block">|e^z|= e^{|\Re (z)|},</math>

where {{tmath|\Re(z)}} denotes the real part of {{tmath|z}}.