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".^{[2]}
When x = π, Euler's formula may be rewritten as e^{iπ} + 1 = 0 or e^{iπ} = -1, which is known as Euler's identity. The version with 0 is considered an example of mathematical beauty due to linking five fundamental mathematical constants with three basic arithmetic operations,^{[3]} each occurring only once.
Historyedit
In 1714, the English mathematician Roger Cotes presented a geometrical argument that can be interpreted (after correcting a misplaced factor of ${\sqrt {-1}}$) as:^{[4]}^{[5]}^{[6]}
$ix=\ln(\cos x+i\sin x).$
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 of 2πi.
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.^{[7]}^{[5]} The formula was first published in 1748 in his foundational work Introductio in analysin infinitorum.^{[8]}
the above equation tells us something about complex 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 understand complex logarithms. 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 exponentiationedit
The exponential function e^{x} for real values of 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 e^{z} for complex values of z simply by substituting z in place of 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 uniqueanalytic continuation of e^{x} to the complex plane.
Here, n is restricted to positive integers, so there is no question about what the power with exponent n means.
Proofsedit
Various proofs of the formula are possible.
Using differentiationedit
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,^{[10]} so this is permitted).^{[11]}
No assumptions are being made about r and 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 e^{ix} is ie^{ix}. Therefore, differentiating both sides gives
Substituting r(cos θ + i sin θ) for e^{ix} and equating real and imaginary parts in this formula gives dr/dx = 0 and dθ/dx = 1. Thus, r is a constant, and θ is x + C for some constant C. The initial values r(0) = 1 and θ(0) = 0 come from e^{0i} = 1, giving r = 1 and θ = x. This proves the formula
This formula can be interpreted as saying that the function e^{iφ} is a unit complex number, i.e., it traces out the unit circle in the complex plane as φ ranges through the real numbers. Here 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 radians.
The original proof is based on the Taylor series expansions of the exponential functione^{z} (where z is a complex number) and of sin x and cos x for real numbers x (see above). In fact, the same proof shows that Euler's formula is even valid for all complex numbers x.
A point in the complex plane can be represented by a complex number written in cartesian coordinates. Euler's formula provides a means of conversion between cartesian coordinates and polar coordinates. The polar form simplifies the mathematics when used in multiplication or powers of complex numbers. Any complex number z = x + iy, and its complex conjugate, z = x − iy, can be written as
φ is the argument of z, i.e., the angle between the x axis and the vector z measured counterclockwise in radians, which is defined up to addition of 2π. Many texts write φ = tan^{−1}y/x instead of φ = atan2(y, x), but the first equation needs adjustment when x ≤ 0. This is because for any real x and y, not both zero, the angles of the vectors (x, y) and (−x, −y) differ by π radians, but have the identical value of tan φ = y/x.
Use of the formula to define the logarithm of complex numbersedit
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):
$a=e^{\ln a},$
and that
$e^{a}e^{b}=e^{a+b},$
both valid for any complex numbers a and b. Therefore, one can write:
for any z ≠ 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 the complex logarithm. The logarithm of a complex number is thus a multi-valued function, because φ is multi-valued.
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 analysis and trigonometry, and provides an interpretation of the sine and cosine functions as weighted sums of the exponential function:
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:
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:
This formula is used for recursive generation of cos nx for integer values of n and arbitrary x (in radians).
Considering cos x a parameter in equation above yields recursive formula for Chebyshev polynomials of the first kind.
Topological interpretationedit
In the language of topology, Euler's formula states that the imaginary exponential function $t\mapsto e^{it}$ is a (surjective) morphism of topological groups from the real line $\mathbb {R}$ to the unit circle $\mathbb {S} ^{1}$. In fact, this exhibits $\mathbb {R}$ as a covering space of $\mathbb {S} ^{1}$. Similarly, Euler's identity says that the kernel of this map is $\tau \mathbb {Z}$, where $\tau =2\pi$. These observations may be combined and summarized in the commutative diagram below:
Other applicationsedit
In differential equations, the function e^{ix} 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.
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 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.
and the element is called a versor in quaternions. The set of all versors forms a 3-sphere in the 4-space.
Other special casesedit
The special 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 at x = τ (where τ = 2π, one turn) yields e^{iτ} = 1 + 0. This is also argued to link five fundamental constants with three basic arithmetic operations, but, unlike Euler's identity, without rearranging the addends from the general case:
^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.
^Gallagher, James (13 February 2014). "Mathematics: Why the brain sees maths as beauty". BBC News online. Retrieved 13 February 2014.
^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 $EX+XC{\sqrt {-1}}$& CE mensura ducta in ${\sqrt {-1}}$." (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 $EX+XC{\sqrt {-1}}$ & CE multiplied by ${\sqrt {-1}}$.) 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 $EX+XC{\sqrt {-1}}$ and CE 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 by θ, which for any angle measured in radians is CE • 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 by CE 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}^{b}John Stillwell (2002). Mathematics and Its History. Springer. ISBN 9781441960528.
^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.
^Hartl, Michael (14 March 2019) [2010-03-14]. "The Tau Manifesto". Archived from the original on 28 June 2019. Retrieved 14 September 2013.
Further readingedit
Nahin, Paul J. (2006). Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills. Princeton University Press. ISBN 978-0-691-11822-2.
Wilson, Robin (2018). Euler's Pioneering Equation: The Most Beautiful Theorem in Mathematics. Oxford: Oxford University Press. ISBN 978-0-19-879492-9. MR 3791469.