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{{short description|Expression of the complex exponential in terms of sine and cosine}}
{{About|Euler's formula in complex analysis|Euler's formula in algebraic topology and polyhedral combinatorics|Euler characteristic}}
{{e (mathematical constant)}}
'''Euler's formula''', named after [[Leonhard Euler]], is a [[mathematics|mathematical]] [[formula]] in [[complex analysis]] that establishes the fundamental relationship between the [[trigonometric functions]] and the [[complex number|complex]] [[exponential function]]. Euler's formula states that for any [[real number]]&nbsp;{{mvar|x}}:

:<math>e^{ix} = \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, with the argument {{mvar|x}} given in [[radian]]s. This complex exponential function is sometimes denoted {{math|[[cis (mathematics)|cis]] ''x''}} ("'''c'''osine plus '''i''' '''s'''ine"). The formula is still valid if {{mvar|x}} is a [[complex number]], and so some authors refer to the more general complex version as Euler's formula.<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}}</ref>

Euler's formula is ubiquitous in mathematics, physics, 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}}</ref>

When <math>x = \pi</math>, Euler's formula evaluates to <math>e^{i \pi} + 1 = 0</math>, which is known as [[Euler's identity]].

==History==
[[Johann Bernoulli]] noted that<ref>{{cite journal|first=Johann |last=Bernoulli |title=Solution d'un problème concernant le calcul intégral, avec quelques abrégés par rapport à ce calcul |trans-title=Solution of a problem in integral calculus with some notes relating to this calculation |journal=Mémoires de l'Académie Royale des Sciences de Paris |pages=197–289 |volume=1702 |date=1702}}</ref>

:<math>\frac{1}{1 + x^2} = \frac12 \left( \frac{1}{1 - ix} + \frac{1}{1 + ix}\right).</math>

And since

:<math>\int \frac{dx}{1 + ax} = \frac{1}{a} \ln(1 + ax) + C,</math>

the above equation tells us something about [[complex logarithm]]s by relating natural logarithms to imaginary (complex) numbers. Bernoulli, however, did not evaluate the integral.

Bernoulli's correspondence with Euler (who also knew the above equation) shows that Bernoulli did not fully understand [[complex logarithm]]s. Euler also suggested that the complex logarithms can have infinitely many values.

Meanwhile, [[Roger Cotes]] in 1714 discovered that<ref name="Stillwell">{{cite book|author=John Stillwell|title=Mathematics and Its History|publisher=Springer|year=2002 | url = https://books.google.com/books?id=V7mxZqjs5yUC&pg=PA315}}</ref>

:<math>ix = \ln(\cos x + i\sin x).</math>

Cotes missed the fact{{cn|reason=The lengthy quote in our bio article makes it clear that Cotes was talking about an arc of a quadrant of a circle, so multiples of pi would not have arisen (except, conceivably, as some sort of mistake in his proof); it also says he didn't use the term natural log, perhaps because it hadn't been invented yet, and used the Latin term for 'measure' instead; at any rate the claim that he 'missed' something, with its connotation of incompetence, seems thoroughly dubious (though not necessarily wrong if his proof was flawed as a result) and in need of a supporting citation from a reliable source, or alternatively some kind of less pejorative and less anachronistic re-wording.|date=January 2018}} that a complex logarithm can have infinitely many values, differing by multiples of {{math|2''i''π}}, due to the periodicity of the trigonometric functions.

Around 1740 Euler turned his attention to the exponential function instead of logarithms and obtained the formula used today that is named after him. It was published in 1748, obtained by comparing the series expansions of the exponential and trigonometric expressions.<ref>[[Leonard Euler]] (1748) [http://www.17centurymaths.com/contents/euler/introductiontoanalysisvolone/ch8vol1.pdf Chapter 8: On transcending quantities arising from the circle] of [[Introduction to the Analysis of the Infinite]], page 214, section 138 (translation by Ian Bruce, pdf link from 17 century maths).</ref><ref name="Stillwell"/>

The view of complex numbers as points in the [[complex plane]] was described about 50 years later by [[Caspar Wessel]].

==Applications in complex number theory==
[[Image:Euler's formula.svg|thumb|right]]
[[Image:Euler's Formula c.png|thumb|upright=1.5|Three-dimensional visualization of Euler's formula. See also [[circular polarization]].]]
;Interpretation of the formula

This formula can be interpreted as saying that the function {{mvar|e<sup>iφ</sup>}} is a [[unit complex number]], i.e., it traces out the [[unit circle]] in the [[complex plane]] as {{mvar|φ}} ranges through the real numbers. Here {{mvar|φ}} is the [[angle]] that a line connecting the origin with a point on the unit circle makes with the [[positive real axis]], measured counterclockwise and in [[radian]]s.

The original proof is based on the [[Taylor series]] expansions of the [[exponential function]] {{mvar|e<sup>z</sup>}} (where {{mvar|z}} is a complex number) and of {{math|sin ''x''}} and {{math|cos ''x''}} for real numbers {{mvar|x}} (see below). In fact, the same proof shows that Euler's formula is even valid for all ''complex'' numbers&nbsp;{{mvar|x}}.

A point in the [[complex plane]] can be represented by a complex number written in [[Coordinates (elementary mathematics)#Cartesian coordinates|cartesian coordinates]]. Euler's formula provides a means of conversion between cartesian coordinates and [[Polar coordinate system|polar coordinates]]. The polar form simplifies the mathematics when used in multiplication or powers of complex numbers. Any complex number {{math|1 = ''z'' = ''x'' + ''iy''}}, and its complex conjugate, {{math|1 = {{overline|''z''}} = ''x'' − ''iy''}}, can be written as

:<math>\begin{align}
z &= x + iy = |z| (\cos \varphi + i\sin \varphi) = r e^{i \varphi}, \\
\bar{z} &= x - iy = |z| (\cos \varphi - i\sin \varphi) = r e^{-i \varphi},
\end{align}</math>

where
:{{math|1=''x'' = Re ''z''}} is the real part,
:{{math|1=''y'' = Im ''z''}} is the imaginary part,
:{{math|1=''r'' = {{abs|''z''}} = {{sqrt|''x''<sup>2</sup> + ''y''<sup>2</sup>}}}} is the [[magnitude (mathematics)|magnitude]] of {{mvar|z}}
:{{math|1=''φ'' = arg ''z'' = [[atan2]](''y'', ''x'')}}.

{{mvar|φ}} is the [[arg (mathematics)|argument]] of {{mvar|z}}, i.e., the angle between the ''x'' axis and the vector ''z'' measured counterclockwise in [[radian]]s, which is defined [[up to]] addition of {{math|2π}}. Many texts write {{math|1=''φ'' = tan<sup>−1</sup> ''{{sfrac|y|x}}''}} instead of {{math|1= ''φ'' = atan2(''y'',''x'')}}, but the first equation needs adjustment when {{mvar|''x'' ≤ 0}}. This is because for any real {{mvar|x}} and {{mvar|y}} not both zero the angles of the vectors {{math|(''x'', ''y'')}} and {{math|(−''x'', −''y'')}} differ by {{math|π}} radians, but have the identical value of {{math|1=tan ''φ'' = ''{{sfrac|y|x}}''}}.

;Use of the formula to define the logarithm of complex numbers

Now, taking this derived formula, we can use Euler's formula to define the [[logarithm]] of a complex number. To do this, we also use the definition of the logarithm (as the inverse operator of exponentiation):

:<math>a = e^{\ln a}, </math>

and that

:<math>e^a e^b = e^{a + b}, </math>

both valid for any complex numbers {{mvar|a}} and {{mvar|b}}.

Therefore, one can write:

:<math>z = |z| e^{i \varphi} = e^{\ln|z|} e^{i \varphi} = e^{\ln|z| + i \varphi}</math>

for any {{math|''z'' ≠ 0}}. Taking the logarithm of both sides shows that

:<math>\ln z = \ln|z| + i \varphi,</math>

and in fact this can be used as the definition for the [[complex logarithm]]. The logarithm of a complex number is thus a [[multi-valued function]], because {{mvar|φ}} is multi-valued.

Finally, the other exponential law

: <math>\left(e^a\right)^k = e^{a k},</math>

which can be seen to hold for all integers {{mvar|k}}, together with Euler's formula, implies several [[trigonometric identity|trigonometric identities]], as well as [[de Moivre's formula]].

==Relationship to trigonometry==
[[File:Sine Cosine Exponential qtl1.svg|thumb|Relationship between sine, cosine and exponential function]]
Euler's formula provides a powerful connection between [[mathematical analysis|analysis]] and [[trigonometry]], and provides an interpretation of the sine and cosine functions as [[weighted sum]]s of the exponential function:

:<math>\begin{align}
\cos x &= \operatorname{Re} \left(e^{ix}\right) =\frac{e^{ix} + e^{-ix}}{2}, \\
\sin x &= \operatorname{Im} \left(e^{ix}\right) =\frac{e^{ix} - e^{-ix}}{2i}.
\end{align}</math>

The two equations above can be derived by adding or subtracting Euler's formulas:

: <math>\begin{align}
e^{ix}  &= \cos x + i \sin x, \\
e^{-ix} &= \cos(- x) + i \sin(- x) = \cos x - i \sin x
\end{align}</math>

and solving for either cosine or sine.

These formulas can even serve as the definition of the trigonometric functions for complex arguments {{mvar|x}}. For example, letting {{math|1=''x'' = ''iy''}}, we have:

:<math>\begin{align}
\cos(iy) &= \frac{e^{-y} + e^y}{2} = \cosh(y), \\
\sin(iy) &= \frac{e^{-y} - e^y}{2i} = i \left(\frac{e^y - e^{-y}}{2}\right) = i\sinh(y).
\end{align}</math>

Complex exponentials can simplify trigonometry, because they are easier to manipulate than their sinusoidal components. One technique is simply to convert sinusoids into equivalent expressions in terms of exponentials. After the manipulations, the simplified result is still real-valued. For example:

:<math>\begin{align}
\cos x\cdot \cos y &= \frac{e^{ix}+e^{-ix}}{2} \cdot \frac{e^{iy}+e^{-iy}}{2} \\
 &= \frac12\cdot \frac{e^{i(x+y)}+e^{i(x-y)}+e^{i(-x+y)}+e^{i(-x-y)}}{2} \\
 &= \frac12 \bigg( \underbrace{ \frac{e^{i(x+y)} + e^{-i(x+y)}}{2} }_{\cos(x+y)} + \underbrace{ \frac{e^{i(x-y)} + e^{-i(x-y)}}{2} }_{\cos(x-y)} \bigg).
\end{align}
</math>

Another technique is to represent the sinusoids in terms of the [[real part]] of a complex expression and perform the manipulations on the complex expression. For example:

:<math>\begin{align}
\cos(nx) &= \operatorname{Re} \left(e^{inx}\right) \\
 &= \operatorname{Re} \left( e^{i(n-1)x}\cdot e^{ix} \right) \\
 &= \operatorname{Re} \Big( e^{i(n-1)x}\cdot \big(\underbrace{e^{ix} + e^{-ix}}_{2\cos x } - e^{-ix}\big) \Big) \\
 &= \operatorname{Re} \left( e^{i(n-1)x}\cdot 2\cos x - e^{i(n-2)x} \right) \\
 &= \cos[(n-1)x] \cdot [2 \cos(x)] - \cos[(n-2)x].
\end{align}</math>

This formula is used for recursive generation of {{math|cos ''nx''}} for integer values of {{mvar|n}} and arbitrary {{mvar|x}} (in radians).

See also [[Phasor#Phasor arithmetic|Phasor arithmetic]].

==Topological interpretation==
In the language of [[topology]], Euler's formula states that the imaginary exponential function {{math|''t'' ↦ ''e<sup>it</sup>''}} is a (surjective) [[morphism]] of [[topological group]]s from the real line {{math|ℝ}} to the unit circle {{math|<math>\mathbb S</math><sup>1</sup>}}. In fact, this exhibits {{math|ℝ}} as a [[covering space]] of <math>\mathbb S^1</math>. Similarly, [[Euler's identity]] says that the [[Kernel (algebra)|kernel]] of this map is {{math|τℤ}}, where {{math|1=τ = 2π}}. These observations may be combined and summarized in the [[commutative diagram]] below:

[[File:Euler's formula commutative diagram.png|frameless|center|Euler's formula and identity combined in diagrammatic form]]

==Other applications==
{{see also|Complex number#Applications}}

In [[differential equations]], the function {{math|''e<sup>ix</sup>''}} is often used to simplify solutions, even if the final answer is a real function involving sine and cosine. The reason for this is that the exponential function is the [[eigenfunction]] of the operation of [[differentiation (mathematics)|differentiation]].

In [[electrical engineering]], [[signal processing]], and similar fields, signals that vary periodically over time are often described as a combination of sinusoidal functions (see [[Fourier analysis]]), and these are more conveniently expressed as the sum of exponential functions with [[imaginary number|imaginary]] exponents, using Euler's formula. Also, [[phasor analysis]] of circuits can include Euler's formula to represent the impedance of a capacitor or an inductor.

In the [[four-dimensional space]] of [[quaternion]]s, there is a [[sphere]] of [[imaginary unit]]s. For any point ''r'' on this sphere, and ''x'' a real number, Euler's formula applies:
:<math>\exp(xr) = \cos x + r \sin x,</math>
and the element is called a [[versor]] in quaternions. The set of all versors forms a [[3-sphere]] in the 4-space.

==Definitions of complex exponentiation==
{{main|Exponentiation|Exponential function}}

The exponential function {{math|''e<sup>x</sup>''}} for real values of {{math|''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 {{math|''z''}} simply by substituting {{math|''z''}} in place of {{math|''x''}} and using the complex algebraic operations. In particular we may use either of the three following definitions, which are equivalent. From a more advanced perspective, each of these definitions may be interpreted as giving the unique [[analytic continuation]] of {{math|''e<sup>x</sup>''}} to the complex plane.

===Differential equation definition===

The exponential function <math>z\mapsto e^z</math> is the unique [[differentiable function]] of a [[complex variable]] such that 
:<math>\frac{de^z}{dz} = e^z</math>
and 
:<math>e^0=1.</math>

===Power series definition===
For complex {{math|''z''}}

:<math>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 {{math|''z''}}.

===Limit definition===
For complex {{math|''z''}}

:<math>e^z = \lim_{n \rightarrow \infty} \left(1+\frac{z}{n}\right)^n.</math>

Here, <i>n</i> is restricted to [[positive integer]]s, so there is no question about what the power with exponent <i>n</i> means.

==Proofs==
Various proofs of the formula are possible.

===Using power series===
Here is a proof of Euler's formula using [[Taylor series|power-series expansions]], as well as basic facts about the powers of {{mvar|i}}:<ref>{{cite book|url=https://books.google.com/books?id=PjK0F0T3NBoC&pg=PA428 |title=A Modern Introduction to Differential Equations |first=Henry J. |last=Ricardo |page=428}}</ref>

: <math>\begin{align}
i^0 &= 1, & i^1 &= i, & i^2 &= -1, & i^3 &= -i, \\
i^4 &= 1, & i^5 &= i, & i^6 &= -1, & i^7 &= -i \\
&\vdots   & &\vdots   & &\vdots    & &\vdots
\end{align}</math>

Using now the power-series definition from above, we see that for real values of {{mvar|x}}

: <math>\begin{align}
 e^{ix} &= 1 + ix + \frac{(ix)^2}{2!} + \frac{(ix)^3}{3!} + \frac{(ix)^4}{4!} + \frac{(ix)^5}{5!} + \frac{(ix)^6}{6!} + \frac{(ix)^7}{7!} + \frac{(ix)^8}{8!} + \cdots \\[8pt]
 &= 1 + ix - \frac{x^2}{2!} - \frac{ix^3}{3!} + \frac{x^4}{4!} + \frac{ix^5}{5!} - \frac{x^6}{6!} - \frac{ix^7}{7!} + \frac{x^8}{8!} + \cdots \\[8pt]
 &= \left( 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!} - \cdots \right) + i\left( x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \right) \\[8pt]
 &= \cos x + i\sin x .
\end{align}</math>

In the last step we have simply recognized the [[Taylor series|Maclaurin series]] for {{math|cos ''x''}} and {{math|sin ''x''}}. The rearrangement of terms is justified because each series is [[absolute convergence|absolutely convergent]].

===Using polar coordinates===
Another proof<ref name=Strang>{{cite book |url=http://ocw.mit.edu/resources/res-18-001-calculus-online-textbook-spring-2005/textbook/ |title=Calculus |first=Gilbert |last=Strang |page=389 |publisher=Wellesley-Cambridge |year=1991 |isbn=0-9614088-2-0}} Second proof on page.</ref> is based on the fact that all complex numbers can be expressed in polar coordinates. Therefore, for ''some'' {{math|''r''}} and {{math|''θ''}} depending on {{math|''x''}},
:<math>e^{ix} = r (\cos \theta + i \sin \theta).</math>
No assumptions are being made about {{mvar|r}} and {{mvar|θ}}; they will be determined in the course of the proof. From any of the definitions of the exponential function it can be shown that the derivative of {{math|''e''<sup>''ix''</sup>}} is {{math|''ie''<sup>''ix''</sup>}}. Therefore, differentiating both sides gives
:<math>i e ^{ix} = (\cos \theta  + i \sin \theta) \frac{dr}{dx} + r (-\sin \theta  + i \cos \theta) \frac{d\theta}{dx}.</math>
Substituting {{math|''r''(cos ''θ'' + ''i'' sin ''θ'')}} for {{math|''e<sup>ix</sup>''}} and equating real and imaginary parts in this formula gives {{math|''{{sfrac|dr|dx}}'' {{=}} 0}} and {{math|''{{sfrac|dθ|dx}}'' {{=}} 1}}. Thus, {{mvar|r}} is a constant, and {{mvar|θ}} is {{math|''x'' + ''C''}} for some constant {{mvar|C}}. The initial values {{math|''r''(0) {{=}} 1}} and {{math|''θ''(0) {{=}} 0}} come from {{math|''e''<sup>0''i''</sup> {{=}} 1}}, giving {{math|''r'' {{=}} 1}} and {{math|''θ'' {{=}} ''x''}}. This proves the formula
:<math>e^{ix} = 1(\cos x +i \sin x) = \cos x + i \sin x.</math>

===Using differential equations===
Another proof is based on [[differential equation]]s satisfied by exponential and trigonometric functions. See {{slink|Trigonometric functions|Relationship to exponential function (Euler's formula)}}.

==See also==
* [[Complex number]]
* [[Euler's identity]]
* [[Integration using Euler's formula]]
* {{section link|History of Lorentz transformations|Euler's gap}}
* [[List of things named after Leonhard Euler]]

==References==
<references/>

==External links==
*[https://web.archive.org/web/20110413234352/http://web.mat.bham.ac.uk/C.J.Sangwin/euler/ Elements of Algebra]
{{Use dmy dates|date=August 2011}}

[[Category:Theorems in complex analysis]]
[[Category:Articles containing proofs]]
[[Category:E (mathematical constant)]]
[[Category:Leonhard Euler]]