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Euler's formula

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Complex exponential in terms of sine and cosine

This article is about Euler's formula in complex analysis. For other uses, seeList of things named after Leonhard Euler § Formulas.

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Euler's formula, named afterLeonhard Euler, is amathematical formula incomplex analysis that establishes the fundamental relationship between thetrigonometric functions and thecomplex exponential function. Euler's formula states that, for anyreal number x, one has $e^{ix}=\cos x+i\sin x,$ wheree is thebase of the natural logarithm,i is theimaginary unit, andcos andsin are thetrigonometric functions cosine andsine respectively. This complex exponential function is sometimes denotedcisx ("cosine plusi sine"). The formula is still valid ifx is acomplex number, and is also calledEuler's formula in this more general case.^[1]

Euler's formula is ubiquitous in mathematics, physics, chemistry, and engineering. The physicistRichard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics".^[2]

Whenx =π, Euler's formula may be rewritten ase^iπ + 1 = 0 ore^iπ = −1, which is known asEuler's identity.

History

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In 1714, the English mathematicianRoger Cotes presented a geometrical argument that can be interpreted (after correcting a misplaced factor of ${\sqrt {-1}}$ ) as:^[3]^[4]^[5] $ix=\ln(\cos x+i\sin x).$ 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 of2πi.

Visualization of Euler's formula as a helix in three-dimensional space. The helix is formed by plotting points for various values of $\theta$ and is determined by both the cosine and sine components of the formula. One curve represents the real component ( $\cos \theta$ ) of the formula, while another curve, rotated 90 degrees around the z-axis (due to multiplication by $i {\displaystyle i}$ ), represents the imaginary component ( $\sin \theta$ ).

{\displaystyle \theta } — Visualization of Euler's formula as a helix in three-dimensional space. The helix is formed by plotting points for various values of $\theta$ and is determined by both the cosine and sine components of the formula. One curve represents the real component ( $\cos \theta$ ) of the formula, while another curve, rotated 90 degrees around the z-axis (due to multiplication by $i {\displaystyle i}$ ), represents the imaginary component ( $\sin \theta$ ).

Around 1740Leonhard 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.^[6]^[4] The formula was first published in 1748 in his foundational workIntroductio in analysin infinitorum.^[7]

Johann Bernoulli had found that^[8] ${\frac {1}{1+x^{2}}}={\frac {1}{2}}\left({\frac {1}{1-ix}}+{\frac {1}{1+ix}}\right).$

And since $\int {\frac {dx}{1+ax}}={\frac {1}{a}}\ln(1+ax)+C,$ the above equation tells us something aboutcomplex logarithms 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 understandcomplex logarithms. Euler also suggested that complex logarithms can have infinitely many values.

The view of complex numbers as points in thecomplex plane was described about 50 years later byCaspar Wessel.

Definitions of complex exponentiation

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Further information:Exponentiation § Complex exponents with a positive real base, andExponential function § On the complex plane

The exponential functione^x for real values ofx may be defined in a few different equivalent ways (seeCharacterizations of the exponential function). Several of these methods may be directly extended to give definitions ofe^z for complex values ofz simply by substitutingz in place ofx 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 theunique analytic continuation ofe^x to the complex plane.

Differential equation definition

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The exponential function $f(z)=e^{z}$ is the uniquedifferentiable function of acomplex variable for which the derivative equals the function ${\frac {df}{dz}}=f$ and $f(0)=1.$

Power series definition

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For complexz $e^{z}=1+{\frac {z}{1!}}+{\frac {z^{2}}{2!}}+{\frac {z^{3}}{3!}}+\cdots =\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}.$

Using theratio test, it is possible to show that thispower series has an infiniteradius of convergence and so definese^z for all complexz.

Limit definition

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For complexz $e^{z}=\lim _{n\to \infty }\left(1+{\frac {z}{n}}\right)^{n}.$

Here,n is restricted topositive integers, so there is no question about what the power with exponentn means.

Proofs

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Various proofs of the formula are possible.

Using differentiation

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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,^[9] so this is permitted).^[10]

Consider the functionf(θ) $f(\theta )={\frac {\cos \theta +i\sin \theta }{e^{i\theta }}}=e^{-i\theta }\left(\cos \theta +i\sin \theta \right)$ for realθ. Differentiating gives by theproduct rule $f'(\theta )=e^{-i\theta }\left(i\cos \theta -\sin \theta \right)-ie^{-i\theta }\left(\cos \theta +i\sin \theta \right)=0$ Thus,f(θ) is a constant. Sincef(0) = 1, thenf(θ) = 1 for all realθ, and thus $e^{i\theta }=\cos \theta +i\sin \theta .$

Using power series

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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 ofe^it fort = 2.

Here is a proof of Euler's formula usingpower-series expansions, as well as basic facts about the powers ofi:^[11] ${\begin{aligned}i^{0}&=1,&i^{1}&=i,&i^{2}&=-1,&i^{3}&=-i,\\i^{4}&=1,&i^{5}&=i,&i^{6}&=-1,&i^{7}&=-i\\&\vdots &&\vdots &&\vdots &&\vdots \end{aligned}}$

Using now the power-series definition from above, we see that for real values ofx ${\begin{aligned}e^{ix}&=1+ix+{\frac {(ix)^{2}}{2!}}+{\frac {(ix)^{3}}{3!}}+{\frac {(ix)^{4}}{4!}}+{\frac {(ix)^{5}}{5!}}+{\frac {(ix)^{6}}{6!}}+{\frac {(ix)^{7}}{7!}}+{\frac {(ix)^{8}}{8!}}+\cdots \\[8pt]&=1+ix-{\frac {x^{2}}{2!}}-{\frac {ix^{3}}{3!}}+{\frac {x^{4}}{4!}}+{\frac {ix^{5}}{5!}}-{\frac {x^{6}}{6!}}-{\frac {ix^{7}}{7!}}+{\frac {x^{8}}{8!}}+\cdots \\[8pt]&=\left(1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+{\frac {x^{8}}{8!}}-\cdots \right)+i\left(x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \right)\\[8pt]&=\cos x+i\sin x,\end{aligned}}$ where in the last step we recognize the two terms are theMaclaurin series forcosx andsinx.The rearrangement of terms is justified because each series isabsolutely convergent.

Using polar coordinates

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Another proof^[12] is based on the fact that all complex numbers can be expressed inpolar coordinates. Therefore,for somer andθ depending onx, $e^{ix}=r\left(\cos \theta +i\sin \theta \right).$ No assumptions are being made aboutr andθ; 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 ofe^ix isie^ix. Therefore, differentiating both sides gives $ie^{ix}=\left(\cos \theta +i\sin \theta \right){\frac {dr}{dx}}+r\left(-\sin \theta +i\cos \theta \right){\frac {d\theta }{dx}}.$ Substitutingr(cosθ +i sinθ) fore^ix and equating real and imaginary parts in this formula gives⁠dr/dx⁠ = 0 and⁠dθ/dx⁠ = 1. Thus,r is a constant, andθ isx +C for some constantC. The initial valuesr(0) = 1 andθ(0) = 0 come frome⁰ⁱ = 1, givingr = 1 andθ =x. This proves the formula $e^{i\theta }=1(\cos \theta +i\sin \theta )=\cos \theta +i\sin \theta .$

Applications

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Applications in complex number theory

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Euler's formulae^iφ = cosφ +i sinφ illustrated in the complex plane.

Interpretation of the formula

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This formula can be interpreted as saying that the functione^iφ is aunit complex number, i.e., it traces out theunit circle in thecomplex plane asφ ranges through the real numbers. Hereφ is theangle that a line connecting the origin with a point on the unit circle makes with thepositive real axis, measured counterclockwise and inradians.

The original proof is based on theTaylor series expansions of theexponential functione^z (wherez is a complex number) and ofsinx andcosx for real numbersx (see above). In fact, the same proof shows that Euler's formula is even valid for allcomplex numbers x.

A point in thecomplex plane can be represented by a complex number written incartesian coordinates. Euler's formula provides a means of conversion between cartesian coordinates andpolar coordinates. The polar form simplifies the mathematics when used in multiplication or powers of complex numbers. Any complex numberz =x +iy, and its complex conjugate,z =x −iy, can be written as ${\begin{aligned}z&=x+iy=|z|(\cos \varphi +i\sin \varphi )=re^{i\varphi },\\{\bar {z}}&=x-iy=|z|(\cos \varphi -i\sin \varphi )=re^{-i\varphi },\end{aligned}}$ where

x = Rez is the real part,
y = Imz is the imaginary part,
r = |z| =√x² +y² is themagnitude ofz and
φ = argz =atan2(y,x).

φ is theargument ofz, i.e., the angle between thex axis and the vectorz measured counterclockwise inradians, which is definedup to addition of2π. Many texts writeφ = tan⁻¹⁠y/x⁠ instead ofφ = atan2(y,x), but the first equation needs adjustment whenx ≤ 0. This is because for any realx andy, not both zero, the angles of the vectors(x,y) and(−x, −y) differ byπ radians, but have the identical value oftanφ =⁠y/x⁠.

Use of the formula to define the logarithm of complex numbers

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Now, taking this derived formula, we can use Euler's formula to define thelogarithm of a complex number. To do this, we also use the definition of the logarithm (as the inverse operator of exponentiation): $a=e^{\ln a},$ and that $e^{a}e^{b}=e^{a+b},$ both valid for any complex numbersa andb. Therefore, one can write: $z=\left|z\right|e^{i\varphi }=e^{\ln \left|z\right|}e^{i\varphi }=e^{\ln \left|z\right|+i\varphi }$ for anyz ≠ 0. Taking the logarithm of both sides shows that $\ln z=\ln \left|z\right|+i\varphi ,$ and in fact, this can be used as the definition for thecomplex logarithm. The logarithm of a complex number is thus amulti-valued function, becauseφ is multi-valued.

Finally, the other exponential law $\left(e^{a}\right)^{k}=e^{ak},$ which can be seen to hold for all integersk, together with Euler's formula, implies severaltrigonometric identities, as well asde Moivre's formula.

Relationship to trigonometry

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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 betweenanalysis andtrigonometry, and provides an interpretation of the sine and cosine functions asweighted sums of the exponential function: ${\begin{aligned}\cos x&=\operatorname {Re} \left(e^{ix}\right)={\frac {e^{ix}+e^{-ix}}{2}},\\\sin x&=\operatorname {Im} \left(e^{ix}\right)={\frac {e^{ix}-e^{-ix}}{2i}}.\end{aligned}}$

The two equations above can be derived by adding or subtracting Euler's formulas: ${\begin{aligned}e^{ix}&=\cos x+i\sin x,\\e^{-ix}&=\cos(-x)+i\sin(-x)=\cos x-i\sin x\end{aligned}}$ and solving for either cosine or sine.

These formulas can even serve as the definition of the trigonometric functions for complex argumentsx. For example, lettingx =iy, we have: ${\begin{aligned}\cos iy&={\frac {e^{-y}+e^{y}}{2}}=\cosh y,\\\sin iy&={\frac {e^{-y}-e^{y}}{2i}}={\frac {e^{y}-e^{-y}}{2}}i=i\sinh y.\end{aligned}}$

In addition ${\begin{aligned}\cosh ix&={\frac {e^{ix}+e^{-ix}}{2}}=\cos x,\\\sinh ix&={\frac {e^{ix}-e^{-ix}}{2}}=i\sin x.\end{aligned}}$

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 calledcomplex sinusoids.^[13] After the manipulations, the simplified result is still real-valued. For example:

${\begin{aligned}\cos x\cos y&={\frac {e^{ix}+e^{-ix}}{2}}\cdot {\frac {e^{iy}+e^{-iy}}{2}}\\&={\frac {1}{2}}\cdot {\frac {e^{i(x+y)}+e^{i(x-y)}+e^{i(-x+y)}+e^{i(-x-y)}}{2}}\\&={\frac {1}{2}}{\bigg (}{\frac {e^{i(x+y)}+e^{-i(x+y)}}{2}}+{\frac {e^{i(x-y)}+e^{-i(x-y)}}{2}}{\bigg )}\\&={\frac {1}{2}}\left(\cos(x+y)+\cos(x-y)\right).\end{aligned}}$

Another technique is to represent sines and cosines in terms of thereal part of a complex expression and perform the manipulations on the complex expression. For example: ${\begin{aligned}\cos nx&=\operatorname {Re} \left(e^{inx}\right)\\&=\operatorname {Re} \left(e^{i(n-1)x}\cdot e^{ix}\right)\\&=\operatorname {Re} {\Big (}e^{i(n-1)x}\cdot {\big (}\underbrace {e^{ix}+e^{-ix}} _{2\cos x}-e^{-ix}{\big )}{\Big )}\\&=\operatorname {Re} \left(e^{i(n-1)x}\cdot 2\cos x-e^{i(n-2)x}\right)\\&=\cos[(n-1)x]\cdot [2\cos x]-\cos[(n-2)x].\end{aligned}}$

This formula is used for recursive generation ofcosnx for integer values ofn and arbitraryx (in radians).

Consideringcosx a parameter in equation above yields recursive formula forChebyshev polynomials of the first kind.

Topological interpretation

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In the language oftopology, Euler's formula states that the imaginary exponential function $t\mapsto e^{it}$ is a (surjective)morphism oftopological groups from the real line $\mathbb {R}$ to the unit circle $\mathbb {S} ^{1}$ . In fact, this exhibits $\mathbb {R}$ as acovering space of $\mathbb {S} ^{1}$ . Similarly,Euler's identity says that thekernel of this map is $\tau \mathbb {Z}$ , where $\tau =2\pi$ . These observations may be combined and summarized in thecommutative diagram below:

Euler's formula and identity combined in diagrammatic form

Other applications

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Other special cases

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Thespecial cases that evaluate to units illustrate rotation around the complex unit circle:

x	e^ix
0 + 2πn	1
⁠π/2⁠ + 2πn	i
π + 2πn	−1
⁠3π/2⁠ + 2πn	−i

The special case atx =τ (whereτ = 2π, oneturn) yieldse^iτ = 1 + 0. This is also argued to link five fundamental constants with three basic arithmetic operations, but, unlike Euler's identity, without rearranging theaddends from the general case: ${\begin{aligned}e^{i\tau }&=\cos \tau +i\sin \tau \\&=1+0\end{aligned}}$ An interpretation of the simplified forme^iτ = 1 is that rotating by a full turn is anidentity function.^[14]

References

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^Moskowitz, Martin A. (2002).A Course in Complex Analysis in One Variable. World Scientific Publishing Co. p. 7.ISBN 981-02-4780-X.
^Feynman, Richard P. (1977).The Feynman Lectures on Physics, vol. I. Addison-Wesley. p. 22-10.ISBN 0-201-02010-6.
^
Cotes wrote:"Nam si quadrantis circuli quilibet arcus, radioCE descriptus, sinun habeatCX sinumque complementi ad quadrantemXE ; sumendo radiumCE pro Modulo, arcus erit rationis inter $EX+XC{\sqrt {-1}}$ &CE mensura ducta in ${\sqrt {-1}}$ ." (Thus if any arc of a quadrant of a circle, described by the radiusCE, has sinusCX and sinus of the complement to the quadrantXE ; taking the radiusCE as modulus, the arc will be the measure of the ratio between $EX+XC{\sqrt {-1}}$ &CE multiplied by ${\sqrt {-1}}$ .) That is, consider a circle having centerE (at the origin of the (x,y) plane) and radiusCE. Consider an angleθ with its vertex atE having the positive x-axis as one side and a radiusCE as the other side. The perpendicular from the pointC on the circle to the x-axis is the "sinus"CX ; the line between the circle's centerE and the pointX at the foot of the perpendicular isXE, which is the "sinus of the complement to the quadrant" or "cosinus". The ratio between $EX+XC{\sqrt {-1}}$ andCE is thus $\cos \theta +{\sqrt {-1}}\sin \theta \$ . 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 ${\sqrt {-1}}$ , equals the length of the circular arc subtended byθ, which for any angle measured in radians isCE •θ. Thus, ${\sqrt {-1}}CE\ln {\left(\cos \theta +{\sqrt {-1}}\sin \theta \right)\ }=(CE)\theta$ . This equation has a misplaced factor: the factor of ${\sqrt {-1}}$ should be on the right side of the equation, not the left side. If the change of scaling by ${\sqrt {-1}}$ is made, then, after dividing both sides byCE and exponentiating both sides, the result is: $\cos \theta +{\sqrt {-1}}\sin \theta =e^{{\sqrt {-1}}\theta }$ , which is Euler's formula.
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:Hathi Trust
- Roger Cotes with Robert Smith, ed.,Harmonia mensurarum … (Cambridge, England: 1722), chapter: "Logometria",p. 28.
- https://nrich.maths.org/1384
^^a ^bJohn Stillwell (2002).Mathematics and Its History. Springer.ISBN 9781441960528.
^Sandifer, C. Edward (2007),Euler's Greatest Hits,Mathematical Association of AmericaISBN 978-0-88385-563-8
^Leonhard Euler (1748)Chapter 8: On transcending quantities arising from the circle ofIntroduction to the Analysis of the Infinite, page 214, section 138 (translation by Ian Bruce, pdf link from 17 century maths).
^Conway & Guy, pp. 254–255
^Bernoulli, Johann (1702). "Solution d'un problème concernant le calcul intégral, avec quelques abrégés par rapport à ce calcul" [Solution of a problem in integral calculus with some notes relating to this calculation].Mémoires de l'Académie Royale des Sciences de Paris.1702:289–297.
^Apostol, Tom (1974).Mathematical Analysis. Pearson. p. 20.ISBN 978-0201002881. Theorem 1.42
^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
^Ricardo, Henry J. (23 March 2016).A Modern Introduction to Differential Equations. Elsevier Science. p. 428.ISBN 9780123859136.
^Strang, Gilbert (1991).Calculus. Wellesley-Cambridge. p. 389.ISBN 0-9614088-2-0. Second proof on page.
^"Complex Sinusoids".ccrma.stanford.edu. Retrieved10 September 2024.
^Hartl, Michael (14 March 2019) [2010-03-14]."The Tau Manifesto".Archived from the original on 28 June 2019. Retrieved14 September 2013.

External links

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Elements of Algebra

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