Editing Exponentiation (section)

===Exponential function===
{{Main|Exponential function}}
The ''exponential function'' may be defined as <math>x\mapsto e^x,</math> where <math>e\approx 2.718</math> is [[Euler's number]], but to avoid [[circular reasoning]], this definition cannot be used here. Rather, we give an independent definition of the exponential function <math>\exp(x),</math> and of <math>e=\exp(1)</math>, relying only on positive integer powers (repeated multiplication). Then we sketch the proof that this agrees with the previous definition: <math>\exp(x)=e^x.</math>

There are [[characterizations of the exponential function|many equivalent ways to define the exponential function]], one of them being
: <math>\exp(x) = \lim_{n\rightarrow\infty} \left(1 + \frac{x}{n}\right)^n.</math>

One has <math>\exp(0)=1,</math> and the ''exponential identity'' (or multiplication rule) <math>\exp(x)\exp(y)=\exp(x+y)</math> holds as well, since
: <math>\exp(x)\exp(y) = \lim_{n\rightarrow\infty} \left(1 + \frac{x}{n}\right)^n\left(1 + \frac{y}{n}\right)^n = \lim_{n\rightarrow\infty} \left(1 + \frac{x+y}{n} + \frac{xy}{n^2}\right)^n,</math>
and the second-order term <math>\frac{xy}{n^2}</math> does not affect the limit, yielding <math>\exp(x)\exp(y) = \exp(x+y)</math>.

Euler's number can be defined as <math>e=\exp(1)</math>. It follows from the preceding equations that <math>\exp(x)=e^x</math> when {{mvar|x}} is an integer (this results from the repeated-multiplication definition of the exponentiation). If {{mvar|x}} is real, <math>\exp(x)=e^x</math> results from the definitions given in preceding sections, by using the exponential identity if {{mvar|x}} is rational, and the continuity of the exponential function otherwise.

The limit that defines the exponential function converges for every [[complex number|complex]] value of {{mvar|x}}, and therefore it can be used to extend the definition of <math>\exp(z)</math>, and thus <math>e^z,</math> from the real numbers to any complex argument {{mvar|z}}. This extended exponential function still satisfies the exponential identity, and is commonly used for defining exponentiation for complex base and exponent.