Editing Euler's formula (section)

{{Short description|Complex exponential in terms of sine and cosine}}
{{About|Euler's formula in complex analysis||List of things named after Leonhard Euler#Formulas}}
{{Use dmy dates|date=October 2021}}
{{e (mathematical constant)}}
'''Euler's formula''', named after [[Leonhard Euler]], is a [[mathematical formula]] in [[complex analysis]] that establishes the fundamental relationship between the [[trigonometric functions]] and the complex [[exponential function]]. Euler's formula states that, for any [[real number]]&nbsp;{{mvar|x}}, one has
<math display="block">e^{i x} = \cos x + i \sin x,  </math>
where {{mvar|e}} is the [[e (mathematical constant)|base of the natural logarithm]], {{mvar|i}} is the [[imaginary unit]], and {{math|cos}} and {{math|sin}} are the [[trigonometric functions]] [[cosine]] and [[sine]] respectively. This complex exponential function is sometimes denoted {{math|[[cis (mathematics)|cis]] ''x''}} ("cosine plus ''i'' sine"). The formula is still valid if {{mvar|x}} is a [[complex number]], and is also called ''Euler's formula'' in this more general case.<ref>{{cite book | first=Martin A. | last= Moskowitz | title=A Course in Complex Analysis in One Variable | publisher = World Scientific Publishing Co. | year=2002 | isbn=981-02-4780-X | pages=7|url={{Google books|Acw5DwAAQBAJ|A Course in Complex Analysis in One Variable|page=7|plainurl=yes}}}}</ref>

Euler's formula is ubiquitous in mathematics, physics, chemistry, and engineering. The physicist [[Richard Feynman]] called the equation "our jewel" and "the most remarkable formula in mathematics".<ref>{{cite book|first=Richard P.|last= Feynman|title=The Feynman Lectures on Physics, vol. I|publisher=Addison-Wesley|year=1977|isbn=0-201-02010-6|page=22-10| url=https://feynmanlectures.caltech.edu/I_22.html#Ch22-S5}}</ref>

When {{math|1=''x'' = ''π''}}, Euler's formula may be rewritten as {{math|1=''e<sup>iπ</sup>'' + 1 = 0}} or {{math|1=''e<sup>iπ</sup>'' = −1}}, which is known as [[Euler's identity]].