Editing E (mathematical constant) (section)

=== Complex numbers ===
The [[exponential function]] {{math|''e''<sup>''x''</sup>}} may be written as a [[Taylor series]]<ref>{{cite book |author-last1=Whittaker |author-first1=Edmund Taylor |author-link1=Edmund Taylor Whittaker |title=A Course Of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions |title-link=A Course of Modern Analysis |author-last2=Watson |author-first2=George Neville |author-link2=George Neville Watson |date=1927-01-02 |publisher=[[Cambridge University Press]] |isbn= |edition=4th |publication-place=Cambridge, UK |page=581}}</ref><ref name="strangherman"></ref>

<math display="block"> e^{x} = 1 + {x \over 1!} + {x^{2} \over 2!} + {x^{3} \over 3!} + \cdots = \sum_{n=0}^{\infty} \frac{x^n}{n!}.</math>

Because this series is [[convergent series|convergent]] for every [[complex number|complex]] value of {{mvar|x}}, it is commonly used to extend the definition of {{math|''e''<sup>''x''</sup>}} to the complex numbers.<ref name="Dennery">{{cite book |first1=P. |last1=Dennery |first2=A. |last2=Krzywicki |title=Mathematics for Physicists |year=1995 |orig-year=1967 |publisher=Dover |isbn=0-486-69193-4 |pages=23–25}}</ref> This, with the Taylor series for [[trigonometric functions|{{math|sin}} and {{math|cos ''x''}}]], allows one to derive [[Euler's formula]]:

<math display="block">e^{ix} = \cos x + i\sin x ,</math>

which holds for every complex {{mvar|x}}.<ref name="Dennery"/> The special case with {{math|''x'' {{=}} [[pi|{{pi}}]]}} is [[Euler's identity]]:

<math display="block">e^{i\pi} + 1 = 0 ,</math>
which is considered to be an exemplar of [[mathematical beauty]] as it shows a profound connection between the most fundamental numbers in mathematics. In addition, it is directly used in [[Lindemann–Weierstrass theorem#Transcendence of e and π|a proof]] that {{pi}} is [[Transcendental number|transcendental]], which implies the impossibility of [[squaring the circle]].<ref>{{cite arXiv|title=The Transcendence of π and the Squaring of the Circle|last1=Milla|first1=Lorenz|eprint=2003.14035|year=2020|class=math.HO }}</ref><ref>{{Cite web|url=https://math.colorado.edu/~rohi1040/expository/eistranscendental.pdf |archive-url=https://web.archive.org/web/20210623215444/https://math.colorado.edu/~rohi1040/expository/eistranscendental.pdf |archive-date=2021-06-23 |url-status=live|title=e is transcendental|last=Hines|first=Robert|website=University of Colorado}}</ref> Moreover, the identity implies that, in the [[principal branch]] of the logarithm,<ref name="Dennery"/>

<math display="block">\ln (-1) = i\pi .</math>

Furthermore, using the laws for exponentiation,

<math display="block">(\cos x + i\sin x)^n = \left(e^{ix}\right)^n = e^{inx} = \cos nx + i \sin nx</math>

for any integer {{mvar|n}}, which is [[de Moivre's formula]].<ref name="Sultan"/>

The expressions of {{math|cos ''x''}} and {{math|sin ''x''}} in terms of the [[exponential function]] can be deduced from the Taylor series:<ref name="Dennery"/>
<math display="block">
  \cos x = \frac{e^{ix} + e^{-ix}}{2} , \qquad
  \sin x = \frac{e^{ix} - e^{-ix}}{2i}.
</math>

The expression
<math display=inline>\cos x + i \sin x</math>
is sometimes abbreviated as {{math|cis(''x'')}}.<ref name="Sultan">{{cite book|title=The Mathematics That Every Secondary School Math Teacher Needs to Know |first1=Alan |last1=Sultan |first2=Alice F. |last2=Artzt |year=2010 |pages=326–328 |publisher=Routledge |isbn=978-0-203-85753-3}}</ref>