Editing Euler's formula (section)

==Definitions of complex exponentiation==
{{further|Exponentiation#Complex exponents with a positive real base|Exponential function#On the complex plane}}

The exponential function {{math|''e<sup>x</sup>''}} for real values of {{mvar|x}} may be defined in a few different equivalent ways (see [[Characterizations of the exponential function]]). Several of these methods may be directly extended to give definitions of {{math|''e<sup>z</sup>''}} for complex values of {{mvar|z}} simply by substituting {{mvar|z}} in place of {{mvar|x}} and using the complex algebraic operations. In particular, we may use any of the three following definitions, which are equivalent. From a more advanced perspective, each of these definitions may be interpreted as giving the [[Identity theorem|unique]] [[analytic continuation]] of {{math|''e<sup>x</sup>''}} to the complex plane.

===Differential equation definition===

The exponential function <math>f(z) = e^z</math> is the unique [[differentiable function]] of a [[complex variable]] for which the derivative equals the function <math display="block">\frac{df}{dz} = f</math> and <math display="block">f(0) = 1.</math>

===Power series definition===
For complex {{mvar|z}}
<math display="block">e^z = 1 + \frac{z}{1!} + \frac{z^2}{2!} + \frac{z^3}{3!} + \cdots = \sum_{n=0}^{\infty} \frac{z^n}{n!}.</math>

Using the [[ratio test]], it is possible to show that this [[power series]] has an infinite [[radius of convergence]] and so defines {{math|''e<sup>z</sup>''}} for all complex {{mvar|z}}.

===Limit definition===
For complex {{mvar|z}}
<math display="block">e^z = \lim_{n \to \infty} \left(1+\frac{z}{n}\right)^n.</math>

Here, {{mvar|n}} is restricted to [[positive integer]]s, so there is no question about what the power with exponent {{mvar|n}} means.