Euler's formula

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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  x:

Contents

where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively. This complex exponential function is sometimes denoted cis x ("cosine plus isine"). The formula is still valid if x is a complex number, and so some authors refer to the more general complex version as Euler's formula. [1]

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". [2]

When x = π, Euler's formula evaluates to e + 1 = 0, which is known as Euler's identity.

History

The English mathematician Roger Cotes (who died in 1716, when Euler was only 9 years old) was the first to know of the formula. [3]

In 1714 he presented a geometrical argument that can be interpreted (after correcting a misplaced factor of ) as: [4] [5]

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 Euler turned his attention to the exponential function instead of logarithms and obtained the formula that is named after him. He obtained the formula by comparing the series expansions of the exponential and trigonometric expressions. [6] [5] It was published in 1748 in the Introductio in analysin infinitorum [7] and Euler may have acquired his knowledge through Swiss compatriot Johann Bernoulli.

Bernoulli noted that [8]

And since

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 the 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

The exponential function ex 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 ez 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 unique analytic continuation of ex to the complex plane.

Differential equation definition

The exponential function is the unique differentiable function of a complex variable such that

and

Power series definition

For complex z

Using the ratio test, it is possible to show that this power series has an infinite radius of convergence and so defines ez for all complex z.

Limit definition

For complex z

Here, n is restricted to positive integers, so there is no question about what the power with exponent 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, [9] so this is permitted). [10]

Let f(θ) be the function

for real θ. Differentiating, we have, by the product rule

Thus, f(θ) is a constant. Since f(0) = 1, then f(θ) = 1 for all real θ, and thus

Using power series

Here is a proof of Euler's formula using power-series expansions, as well as basic facts about the powers of i: [11]

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

where in the last step we recognize the two terms are the Maclaurin series for cos x and sin x. The rearrangement of terms is justified because each series is absolutely convergent.

Using polar coordinates

Another proof [12] is based on the fact that all complex numbers can be expressed in polar coordinates. Therefore, for somer and θ depending on x,

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 eix is ieix. Therefore, differentiating both sides gives

Substituting r(cos θ + i sin θ) for eix and equating real and imaginary parts in this formula gives dr/dx = 0 and /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 e0i = 1, giving r = 1 and θ = x. This proves the formula

Applications

Applications in complex number theory

Euler's formula e = cos ph + i sin ph illustrated in the complex plane. Euler's formula.svg
Euler's formula e = cos φ + i sin φ illustrated in the complex plane.
Three-dimensional visualization of Euler's formula. See also circular polarization. Euler's Formula c.png
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 e 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 function ez (where z is a complex number) and of sin x and cos x for real numbers x (see below). 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 = xiy, can be written as

where

x = Re z is the real part,
y = Im z is the imaginary part,
r = |z| = x2 + y2 is the magnitude of z and
φ = arg z = atan2(y, x).

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−1y/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 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):

and that

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

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.

Finally, the other exponential law

which can be seen to hold for all integers k, together with Euler's formula, implies several trigonometric identities, as well as de Moivre's formula.

Relationship to trigonometry

Relationship between sine, cosine and exponential function Sine Cosine Exponential qtl1.svg
Relationship between sine, cosine and exponential function

Euler's formula 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:

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

and solving for either cosine or sine.

These formulas can even serve as the definition of the trigonometric functions for complex arguments x. For example, letting x = iy, we have:

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).

See also Phasor arithmetic.

Topological interpretation

In the language of topology, Euler's formula states that the imaginary exponential function is a (surjective) morphism of topological groups from the real line to the unit circle . In fact, this exhibits as a covering space of . Similarly, Euler's identity says that the kernel of this map is , where . These observations may be combined and summarized in the commutative diagram below:

Euler's formula and identity combined in diagrammatic form Euler's formula commutative diagram.png
Euler's formula and identity combined in diagrammatic form

Other applications

In differential equations, the function eix 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.

In the four-dimensional space of quaternions, there is a sphere of imaginary units. For any point r on this sphere, and x a real number, Euler's formula applies:

and the element is called a versor in quaternions. The set of all versors forms a 3-sphere in the 4-space.

See also

Related Research Articles

Complex number Element of a number system in which –1 has a square root

In mathematics, a complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a symbol called the imaginary unit, and satisfying the equation i2 = −1. Because no "real" number satisfies this equation, i was called an imaginary number by René Descartes. For the complex number a + bi, a is called the real part and b is called the imaginary part. The set of complex numbers is denoted by either of the symbols or C. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world.

Logarithm Inverse of the exponential function, which maps products to sums

In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x. In the simplest case, the logarithm counts the number of occurrences of the same factor in repeated multiplication; e.g., since 1000 = 10 × 10 × 10 = 103, the "logarithm base 10" of 1000 is 3, or log10(1000) = 3. The logarithm of x to baseb is denoted as logb(x), or without parentheses, logbx, or even without the explicit base, log x, when no confusion is possible, or when the base does not matter such as in big O notation.

Polar coordinate system Two-dimensional coordinate system where each point is determined by a distance from reference point and an angle from a reference direction

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or azimuth. The radial coordinate is often denoted by r or ρ, and the angular coordinate by φ, θ, or t. Angles in polar notation are generally expressed in either degrees or radians.

Spherical coordinate system 3-dimensional coordinate system

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Inverse trigonometric functions arcsin, arccos, arctan, etc.

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In probability theory, the Borel–Kolmogorov paradox is a paradox relating to conditional probability with respect to an event of probability zero. It is named after Émile Borel and Andrey Kolmogorov.

Theta function Special functions of several complex variables

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Tangent half-angle formula Relates the tangent of half of an angle to trigonometric functions of the entire angle

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Multiple integral Generalization of definite integrals to functions of multiple variables

In mathematics, a multiple integral is a definite integral of a function of several real variables, for instance, f(x, y) or f(x, y, z). Integrals of a function of two variables over a region in are called double integrals, and integrals of a function of three variables over a region in are called triple integrals. For multiple integrals of a single-variable function, see the Cauchy formula for repeated integration.

Sine trigonometric function of an angle

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The main trigonometric identities between trigonometric functions are proved, using mainly the geometry of the right triangle. For greater and negative angles, see Trigonometric functions.

In mathematics, vector spherical harmonics (VSH) are an extension of the scalar spherical harmonics for use with vector fields. The components of the VSH are complex-valued functions expressed in the spherical coordinate basis vectors.

The trigonometric functions for real or complex square matrices occur in solutions of second-order systems of differential equations. They are defined by the same Taylor series that hold for the trigonometric functions of real and complex numbers:

References

  1. Moskowitz, Martin A. (2002). A Course in Complex Analysis in One Variable. World Scientific Publishing Co. p. 7. ISBN   981-02-4780-X.
  2. Feynman, Richard P. (1977). The Feynman Lectures on Physics, vol. I. Addison-Wesley. p. 22-10. ISBN   0-201-02010-6.
  3. Sandifer, C. Edward (2007), Euler's Greatest Hits , Mathematical Association of America ISBN   978-0-88385-563-8
  4. 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 & CE mensura ducta in ." (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 & CE multiplied by .) 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 and CE is thus . 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 , equals the length of the circular arc subtended by θ, which for any angle measured in radians is CEθ. Thus, . This equation has the wrong sign: the factor of should be on the right side of the equation, not the left side. If this change is made, then, after dividing both sides by CE and exponentiating both sides, the result is: , 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.
  5. 1 2 John Stillwell (2002). Mathematics and Its History. Springer.
  6. Leonard Euler (1748) 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).
  7. Conway & Guy, p. 254–255
  8. 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.
  9. Apostol, Tom (1974). Mathematical Analysis. Pearson. p. 20. ISBN   978-0201002881. Theorem 1.42
  10. 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
  11. Ricardo, Henry J. A Modern Introduction to Differential Equations. p. 428.
  12. Strang, Gilbert (1991). Calculus. Wellesley-Cambridge. p. 389. ISBN   0-9614088-2-0. Second proof on page.

Further reading