Editing Euler's formula

{{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]].

==History==
In 1714, the English mathematician [[Roger Cotes]] presented a geometrical argument that can be interpreted (after correcting a misplaced factor of <math>\sqrt{-1}</math>) as:<ref>Cotes wrote:  ''"Nam si quadrantis circuli quilibet arcus, radio ''CE'' descriptus, sinun habeat ''CX'' sinumque complementi ad quadrantem ''XE'' ; sumendo radium ''CE'' pro Modulo, arcus erit rationis inter <math>EX + XC \sqrt{-1}</math>& ''CE'' mensura ducta in <math>\sqrt{-1}</math>."'' (Thus if any arc of a quadrant of a circle, described by the radius ''CE'', has sinus ''CX'' and sinus of the complement to the quadrant ''XE'' ; taking the radius ''CE'' as modulus, the arc will be the measure of the ratio between <math>EX + XC \sqrt{-1}</math> & ''CE'' multiplied by <math>\sqrt{-1}</math>.) That is, consider a circle having center ''E'' (at the origin of the (x,y) plane) and radius ''CE''.  Consider an angle ''θ'' with its vertex at ''E'' having the positive x-axis as one side and a radius ''CE'' as the other side.  The perpendicular from the point ''C'' on the circle to the x-axis is the "sinus" ''CX'' ; the line between the circle's center ''E'' and the point ''X'' at the foot of the perpendicular is ''XE'', which is the "sinus of the complement to the quadrant" or "cosinus".  The ratio between <math>EX + XC \sqrt{-1}</math> and ''CE'' is thus <math>\cos \theta + \sqrt{-1} \sin \theta \ </math>.  In Cotes' terminology, the "measure" of a quantity is its natural logarithm, and the "modulus" is a conversion factor that transforms a measure of angle into circular arc length (here, the modulus is the radius (''CE'') of the circle).  According to Cotes, the product of the modulus and the measure (logarithm) of the ratio, when multiplied by <math>\sqrt{-1}</math>, equals the length of the circular arc subtended by ''θ'', which for any angle measured in radians is ''CE'' • ''θ''.  Thus, <math>\sqrt{-1} CE \ln{\left ( \cos \theta + \sqrt{-1} \sin \theta \right ) \ } = (CE) \theta </math>.  This equation has a misplaced factor:  the factor of <math>\sqrt{-1}</math> should be on the right side of the equation, not the left side.  If the change of scaling by <math>\sqrt{-1}</math> is made, then, after dividing both sides by ''CE'' and exponentiating both sides, the result is:  <math>\cos \theta + \sqrt{-1} \sin \theta = e^{\sqrt{-1} \theta}</math>, which is Euler's formula.<br />
See:

* Roger Cotes (1714) "Logometria," ''Philosophical Transactions of the Royal Society of London'', '''29''' (338) : 5-45; see especially page 32.  Available on-line at:  [http://babel.hathitrust.org/cgi/pt?id=ucm.5324351035;view=2up;seq=38 Hathi Trust]
* Roger Cotes with Robert Smith, ed., ''Harmonia mensurarum'' … (Cambridge, England:  1722), chapter:  "Logometria",  [https://books.google.com/books?id=J6BGAAAAcAAJ&pg=PA28 p. 28].
* https://nrich.maths.org/1384</ref><ref name="Stillwell">{{cite book|author=John Stillwell|title=Mathematics and Its History|publisher=Springer|year=2002 |isbn=9781441960528| url = https://books.google.com/books?id=V7mxZqjs5yUC&pg=PA315}}</ref><ref>Sandifer, C. Edward (2007), ''[https://books.google.com/books?id=sohHs7ExOsYC&pg=PA4 Euler's Greatest Hits]'', [[Mathematical Association of America]] {{ISBN|978-0-88385-563-8}}</ref>
<math display="block">ix = \ln(\cos x + i\sin x).</math>
Exponentiating this equation yields Euler's formula. Note that the logarithmic statement is not universally correct for complex numbers, since a complex logarithm can have infinitely many values, differing by multiples of {{math|2''πi''}}.
[[File:Rising circular.gif|thumb|Visualization of Euler's formula as a helix in three-dimensional space. The helix is formed by plotting points for various values of <math>\theta</math> and is determined by both the cosine and sine components of the formula. One curve represents the real component (<math>\cos\theta</math>) of the formula, while another curve, rotated 90 degrees around the z-axis (due to multiplication by <math>i</math>), represents the imaginary component (<math>\sin\theta</math>).]]
Around 1740 [[Leonhard Euler]] turned his attention to the exponential function and derived the equation named after him by comparing the series expansions of the exponential and trigonometric expressions.<ref>[[Leonhard 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 formula was first published in 1748 in his foundational work ''[[Introductio in analysin infinitorum]]''.<ref>Conway & Guy, pp. 254–255</ref>

[[Johann Bernoulli]] had found 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=289–297|volume=1702 |date=1702}}</ref>
<math display="block">\frac{1}{1 + x^2} = \frac 1 2 \left( \frac{1}{1 - ix} + \frac{1}{1 + ix}\right).</math>

And since
<math display="block">\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 complex logarithms can have infinitely many values.

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

==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.

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

===Using differentiation===
This proof shows that the quotient of the trigonometric and exponential expressions is the constant function one, so they must be equal (the exponential function is never zero,<ref name=Apostol>{{cite book |last=Apostol |first=Tom |title=Mathematical Analysis |page=20 |publisher=Pearson |date=1974 |isbn=978-0201002881}} Theorem 1.42</ref> so this is permitted).<ref>user02138 (https://math.stackexchange.com/users/2720/user02138), How to prove Euler's formula: $e^{i\varphi}=\cos(\varphi) +i\sin(\varphi)$?, URL (version: 2018-06-25): https://math.stackexchange.com/q/8612</ref>

Consider the function {{math|''f''(''θ'')}} 
<math display="block"> f(\theta) = \frac{\cos\theta + i\sin\theta}{e^{i\theta}} = e^{-i\theta} \left(\cos\theta + i \sin\theta\right) </math>
for real {{mvar|θ}}. Differentiating gives by the [[product rule]]
<math display="block"> f'(\theta) = e^{-i\theta} \left(i\cos\theta - \sin\theta\right) - ie^{-i\theta} \left(\cos\theta + i\sin\theta\right) = 0</math>
Thus, {{math|''f''(''θ'')}} is a constant. Since {{math|1=''f''(0) = 1}}, then {{math|1=''f''(''θ'') = 1}} for all real {{mvar|θ}}, and thus 
<math display="block">e^{i\theta} = \cos\theta + i\sin\theta.</math>

===Using power series===
[[File:Eulers-forrmula-standalone.svg|thumb|alt=Each successive term in the series rotates 90 degrees counter clockwise. The even-power terms are real, hence parallel to the real line, and the odd-power terms are imaginary, hence parallel to the imaginary axis. Plotting each term as a vectors in the complex plane lying end-to-end (vector addition) results in a piecewise-linear spiral starting from the origin and converging to the point (cos 2, sin 2) on the unit circle. |A plot of the first few terms of the Taylor series of {{math|''e''<sup>''it''</sup>}} for {{math|''t'' {{=}} 2}}. ]]

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 |date=23 March 2016 |page=428|publisher=Elsevier Science |isbn=9780123859136 }}</ref>
<math display="block">\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 display="block">\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>
where in the last step we recognize the two terms are the [[Taylor series#Trigonometric functions|Maclaurin series]] for {{math|cos ''x''}} and {{math|sin ''x''}}. [[Riemann series theorem|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]] {{mvar|r}} and {{mvar|θ}} depending on {{mvar|x}},
<math display="block">e^{i x} = r \left(\cos \theta + i \sin \theta\right).</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 display="block">i e ^{ix} = \left(\cos \theta  + i \sin \theta\right) \frac{dr}{dx} + r \left(-\sin \theta + i \cos \theta\right) \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|1=''{{sfrac|dr|dx}}'' = 0}} and {{math|1=''{{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|1=''r''(0) = 1}} and {{math|1=''θ''(0) = 0}} come from {{math|1=''e''<sup>0''i''</sup> = 1}}, giving {{math|1=''r'' = 1}} and {{math|1=''θ'' = ''x''}}. This proves the formula
<math display="block">e^{i \theta} = 1(\cos \theta +i  \sin \theta) = \cos \theta + i  \sin \theta.</math>

==Applications==

===Applications in complex number theory===
[[Image:Euler's formula.svg|thumb|right|Euler's formula {{math|1=''e<sup>iφ</sup>'' = cos ''φ'' + ''i'' sin ''φ''}} illustrated in the complex plane.]]

==== Interpretation of the formula ====
This formula can be interpreted as saying that the function {{math|''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]] {{math|''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}} ([[Euler's formula#Using power series|see above]]). 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 display="block">\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}} and
*{{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|2''π''}}. 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 {{math|''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 {{pi}} 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 display="block">a = e^{\ln a}, </math>
and that
<math display="block">e^a e^b = e^{a + b}, </math>
both valid for any complex numbers {{mvar|a}} and {{mvar|b}}. Therefore, one can write:
<math display="block">z = \left|z\right| e^{i \varphi} = e^{\ln\left|z\right|} e^{i \varphi} = e^{\ln\left|z\right| + i \varphi}</math>
for any {{math|''z'' ≠ 0}}. Taking the logarithm of both sides shows that
<math display="block">\ln z = \ln \left|z\right| + 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 display="block">\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 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, the definitions of the trigonometric functions and the standard identities for exponentials are sufficient to easily derive most trigonometric identities. It 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 display="block">\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 display="block">\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 display="block">\begin{align}
\cos iy &= \frac{e^{-y} + e^y}{2} = \cosh y, \\
\sin iy &= \frac{e^{-y} - e^y}{2i} = \frac{e^y - e^{-y}}{2}i = i\sinh y.
\end{align}</math>

In addition
<math display="block">\begin{align}
\cosh ix &= \frac{e^{ix} + e^{-ix}}{2} = \cos x, \\
\sinh ix &= \frac{e^{ix} - e^{-ix}}{2} = i\sin x.
\end{align}</math>

Complex exponentials can simplify trigonometry, because they are mathematically easier to manipulate than their sine and cosine components. One technique is simply to convert sines and cosines into equivalent expressions in terms of exponentials sometimes called ''complex sinusoids''.<ref>{{Cite web |title=Complex Sinusoids |url=https://ccrma.stanford.edu/~jos/filters06/Complex_Sinusoids.html |access-date=2024-09-10 |website=ccrma.stanford.edu}}</ref> After the manipulations, the simplified result is still real-valued. For example:

<math display="block">\begin{align}
\cos x \cos y &= \frac{e^{ix}+e^{-ix}}{2} \cdot \frac{e^{iy}+e^{-iy}}{2} \\
 &= \frac 1 2 \cdot \frac{e^{i(x+y)}+e^{i(x-y)}+e^{i(-x+y)}+e^{i(-x-y)}}{2} \\
 &= \frac 1 2 \bigg( \frac{e^{i(x+y)} + e^{-i(x+y)}}{2} + \frac{e^{i(x-y)} + e^{-i(x-y)}}{2} \bigg)\\
 &= \frac 1 2 \left( \cos(x+y) + \cos(x-y) \right).
\end{align}
</math>

Another technique is to represent sines and cosines in terms of the [[real part]] of a complex expression and perform the manipulations on the complex expression. For example:
<math display="block">\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).

Considering {{math|cos ''x''}} a parameter in equation above yields recursive formula for [[Chebyshev polynomials]] of the first kind.

{{see also|Phasor#Arithmetic}}

===Topological interpretation===
{{Unreferenced section|date=November 2022}}
In the language of [[topology]], Euler's formula states that the imaginary exponential function <math>t \mapsto e^{it}</math> is a ([[Surjective function|surjective]]) [[morphism]] of [[topological group]]s from the real line <math>\mathbb R</math> to the unit circle <math>\mathbb S^1</math>. In fact, this exhibits <math>\mathbb R</math> 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>\tau \mathbb Z</math>, where <math>\tau = 2\pi</math>. These observations may be combined and summarized in the [[commutative diagram]] below:

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

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

In [[differential equation]]s, 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 {{mvar|r}} on this sphere, and {{mvar|x}} a real number, Euler's formula applies:
<math display="block">\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.

==Other special cases==
The [[special case]]s that evaluate to units illustrate rotation around the complex unit circle:

{|class="wikitable" style="text-align:right;"
! {{math|''x''}} !! {{math|1=''e<sup>ix</sup>''}}
|-
| {{math|0 + 2''πn''}} || {{math|1}}
|-
| {{math|{{sfrac|''π''|2}} + 2''πn''}} || {{math|''i''}}
|-
| {{math|''π'' + 2''πn''}} || {{math|−1}}
|-
| {{math|{{sfrac|''3π''|2}} + 2''πn''}} || {{math|−''i''}}
|}

The special case at {{math|1=''x'' = ''τ''}} (where {{math|1=''τ'' = 2''π''}}, one [[Turn (angle)|turn]]) yields {{math|1=''e<sup>iτ</sup>'' = 1 + 0}}. This is also argued to link five fundamental constants with three basic arithmetic operations, but, unlike Euler's identity, without rearranging the [[addend]]s from the general case:
<math display="block">\begin{align}
e^{i\tau} &= \cos \tau + i \sin \tau \\
&= 1 + 0
\end{align}</math>
An interpretation of the simplified form {{math|1=''e<sup>iτ</sup>'' = 1}} is that rotating by a full turn is an [[identity function]].<ref name="Hartl_2019">{{cite web |title=The Tau Manifesto |author-first=Michael |author-last=Hartl |author-link=Michael Hartl |date=2019-03-14 |orig-date=2010-03-14 |url=http://tauday.com/tau-manifesto |access-date=2013-09-14 |url-status=live |archive-url=https://web.archive.org/web/20190628230418/https://tauday.com/tau-manifesto |archive-date=2019-06-28}}</ref>

==See also==
* [[Complex number]]
* [[Euler's identity]]
* [[Integration using Euler's formula]]
* [[History of Lorentz transformations]]
* [[List of topics named after Leonhard Euler]]

==References==
<references/>

==Further reading==
* {{cite book |first=Paul J. |last=Nahin |title=Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills | url=https://archive.org/details/dreulersfabulous0000nahi |url-access=registration |year=2006 |publisher=Princeton University Press |isbn=978-0-691-11822-2}}
* {{cite book | last = Wilson | first = Robin | isbn = 978-0-19-879492-9 | mr = 3791469 | publisher = Oxford University Press | location = Oxford | title = Euler's Pioneering Equation: The Most Beautiful Theorem in Mathematics | year = 2018}}

==External links==
*[https://web.archive.org/web/20110413234352/http://web.mat.bham.ac.uk/C.J.Sangwin/euler/ Elements of Algebra]

{{Leonhard Euler}}

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