Editing Complex number (section)

===Complex exponential===
[[File:ComplexExpMapping.svg|thumb|right|Illustration of the complex exponential function mapping the complex plane, ''w'' = exp ⁡(''z''). The left plane shows a square mesh with mesh size 1, with the three complex numbers 0, 1, and ''i'' highlighted. The two rectangles (in magenta and green) are mapped to circular segments, while the lines parallel to the ''x''-axis are mapped to rays emanating from, but not containing the origin. Lines parallel to the ''y''-axis are mapped to circles.]]
Like in real analysis, this notion of convergence is used to construct a number of [[elementary function]]s: the ''[[exponential function]]'' {{math|exp ''z''}}, also written {{math|''e''<sup>''z''</sup>}}, is defined as the [[infinite series]], which can be shown to [[radius of convergence|converge]] for any ''z'':
<math display=block>\exp z:= 1+z+\frac{z^2}{2\cdot 1}+\frac{z^3}{3\cdot 2\cdot 1}+\cdots = \sum_{n=0}^{\infty} \frac{z^n}{n!}. </math>
For example, <math>\exp (1)</math> is [[E (mathematical constant)|Euler's number]] <math>e \approx 2.718</math>. ''[[Euler's formula]]'' states:
<math display=block>\exp(i\varphi) = \cos \varphi + i\sin \varphi </math>
for any real number {{mvar|φ}}. This formula is a quick consequence of general basic facts about convergent power series and the definitions of the involved functions as power series. As a special case, this includes [[Euler's identity]]
<math display=block>\exp(i \pi) = -1. </math>