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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:

- History
- Definitions of complex exponentiation
- Differential equation definition
- Power series definition
- Limit definition
- Proofs
- Using differentiation
- Using power series
- Using polar coordinates
- Applications
- Applications in complex number theory
- Interpretation of the formula
- Use of the formula to define the logarithm of complex numbers
- Relationship to trigonometry
- Topological interpretation
- Other applications
- See also
- References
- Further reading
- External links

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* ("**c**osine plus **i****s**ine"). 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 ^{iπ}* + 1 = 0, which is known as Euler's identity.

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 2*πi*.

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.

The exponential function *e ^{x}* for real values of

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

and

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 *e ^{z}* for all complex

For complex z

Here, n is restricted to positive integers, so there is no question about what the power with exponent n means.

Various proofs of the formula are possible.

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

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.

Another proof^{ [12] } is based on the fact that all complex numbers can be expressed in polar coordinates. Therefore, for *some**r* 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 *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

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 function *e*^{z} (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* = *x* − *iy*, can be written as

where

*x*= Re*z*is the real part,*y*= Im*z*is the imaginary part,*r*= |*z*| = √*x*^{2}+*y*^{2}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 2*π*. 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*.

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.

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.

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:

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.

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.

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 *i*^{2} = −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.

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 = 10^{3}, the "logarithm base 10" of 1000 is 3, or log_{10}(1000) = 3. The logarithm of x to *base*b is denoted as log_{b}(*x*), or without parentheses, log_{b} *x*, 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.

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.

In mathematics, a **spherical coordinate system** is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the *radial distance* of that point from a fixed origin, its *polar angle* measured from a fixed zenith direction, and the *azimuthal angle* of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system.

In mathematics, the **trigonometric functions** are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.

In mathematics, **hyperbolic functions** are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos *t*, sin *t*) form a circle with a unit radius, the points (cosh *t*, sinh *t*) form the right half of the unit hyperbola. Also, just as the derivatives of sin(*t*) and cos(*t*) are cos(*t*) and –sin(*t*), the derivatives of sinh(*t*) and cosh(*t*) are cosh(*t*) and +sinh(*t*).

In mathematics, **de Moivre's formula ** states that for any real number x and integer n it holds that

In mathematics and physical science, **spherical harmonics** are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.

In mathematics, the **inverse trigonometric functions** are the inverse functions of the trigonometric functions. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.

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.

In mathematics, **theta functions** are special functions of several complex variables. They are important in many areas, including the theories of Abelian varieties and moduli spaces, and of quadratic forms. They have also been applied to soliton theory. When generalized to a Grassmann algebra, they also appear in quantum field theory.

In trigonometry, **tangent half-angle formulas** relate the tangent of half of an angle to trigonometric functions of the entire angle. The tangent of half an angle is the stereographic projection of the circle onto a line. Among these formulas are the following:

This is a **table of orthonormalized spherical harmonics** that employ the Condon-Shortley phase up to degree . Some of these formulas give the "Cartesian" version. This assumes *x*, *y*, *z*, and *r* are related to and through the usual spherical-to-Cartesian coordinate transformation:

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.

In mathematics, the **sine** is a trigonometric function of an angle. The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle, to the length of the longest side of the triangle. For an angle , the sine function is denoted simply as .

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:

- ↑ 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. - ↑ Sandifer, C. Edward (2007),
*Euler's Greatest Hits*, Mathematical Association of America ISBN 978-0-88385-563-8 - ↑ 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*CE*•*θ*. 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.

- Roger Cotes (1714) "Logometria,"
- 1 2 John Stillwell (2002).
*Mathematics and Its History*. Springer. - ↑ 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).
- ↑ Conway & Guy, p. 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.
*A Modern Introduction to Differential Equations*. p. 428. - ↑ Strang, Gilbert (1991).
*Calculus*. Wellesley-Cambridge. p. 389. ISBN 0-9614088-2-0. Second proof on page.

- 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.

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