Euler\'s formula - Citizendia

This article is about Euler's formula in complex analysis. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex For Euler's formula in graph theory and polyhedral combinatorics see Euler characteristic. In Mathematics, and more specifically in Algebraic topology and Polyhedral combinatorics, the Euler characteristic is a Topological invariant See also topics named after Euler. In Mathematics and Physics, there are a large number of topics named in honour of Leonhard Euler ( pronounced '''''Oiler''''')

Part of a series of articles on
The mathematical constant, e

Natural logarithm

Applications in Compound interest · Euler's identity & Euler's formula · Half-lives & Exponential growth/decay

Defining e Proof that e is irrational · Representations of e · Lindemann–Weierstrass theorem

People John Napier · Leonhard Euler

Schanuel's conjecture

Euler's formula states that, for any real number x,

$e^{ix} = \cos(x) + i\sin(x) \!$

where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are trigonometric functions (here it is assumed that, when calculating the sine and cosine, x is measured in radians rather than in degrees). The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line The natural logarithm, formerly known as the Hyperbolic logarithm is the Logarithm to the base e, where e is an irrational Compound interest is the concept of adding accumulated Interest back to the principal so that interest is earned on interest from that moment on In Mathematical analysis, Euler's identity, named after Leonhard Euler, is the equation e^{i \pi} + 1 = 0 \\! where Half-Life (computer-game page here It's already listed in the disambiguation page Exponential growth (including Exponential decay) occurs when the growth rate of a mathematical function is proportional to the function's current value A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value In Mathematics, the series representation of Euler's number e e = \sum_{n = 0}^{\infty} \frac{1}{n!}\! can be used to prove The Mathematical constant ''e'' can be represented in a variety of ways as a Real number. In Mathematics, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers For other people with the same name see John Napier (disambiguation. In Mathematics, specifically Transcendence theory, Schanuel's conjecture is the following statement Given any n Complex numbers In Mathematics, the real numbers may be described informally in several different ways The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation The radian is a unit of plane Angle, equal to 180/ π degrees, or about 57 This article describes the unit of angle For other meanings see Degree. 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. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted ^[1]

Richard Feynman called Euler's formula "our jewel" and "the most remarkable formula in mathematics". Richard Phillips Feynman (ˈfaɪnmən May 11 1918 – February 15 1988 was an American Physicist known for the Path integral formulation of quantum ^[2]

1 History
2 Applications in complex number theory
3 Relationship to trigonometry
4 Other applications
5 Definitions of complex exponentiation
6 Proofs
7 See also
8 References
9 External links

History

Euler's formula was proven for the first time by Roger Cotes in 1714 in the form

$\ln(\cos(x) + i\sin(x))=ix \$

(where "ln" means natural logarithm, i. In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true Roger Cotes FRS ( July 10, 1682 – June 5, 1716) was an English Mathematician, known for working closely with Year 1714 ( MDCCXIV) was a Common year starting on Monday (link will display the full calendar of the Gregorian calendar (or a The natural logarithm, formerly known as the Hyperbolic logarithm is the Logarithm to the base e, where e is an irrational e. log with base e). ^[3]

It was Euler who published the equation in its current form in 1748, basing his proof on the infinite series of both sides being equal. Year 1748 ( MDCCXLVIII) was a Leap year starting on Monday (link will display the full calendar of the Gregorian calendar (or a In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with Neither of these men saw the geometrical interpretation of the formula: the view of complex numbers as points in the complex plane arose only some 50 years later (see Caspar Wessel). In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis Caspar Wessel ( June 8, 1745 - March 25, 1818) was a Danish-Norwegian Mathematician. Euler considered it natural to introduce students to complex numbers much earlier than we do today. In his elementary algebra text book, Elements of Algebra, he introduces these numbers almost at once and then uses them in a natural way throughout.

Applications in complex number theory

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics and in the Sciences a formula (plural formulae, formulæ or formulas) is a concise way of expressing information Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) (Euler's identity is a special case of the Euler formula. In Mathematical analysis, Euler's identity, named after Leonhard Euler, is the equation e^{i \pi} + 1 = 0 \\! where )

This formula can be interpreted as saying that the function e^ix traces out the unit circle in the complex number plane as x ranges through the real numbers. In Mathematics, a unit circle is Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Here, x is the angle that a line connecting the origin with a point on the unit circle makes with the positive real axis, measured counter clockwise and in radians. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called The radian is a unit of plane Angle, equal to 180/ π degrees, or about 57

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 Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) In fact, the same proof shows that Euler's formula is even valid for all complex numbers z.

A point in the complex plane can be represented by a complex number written in cartesian coordinates. In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis Coordinates are numbers which describe the location of points in a plane or in space Euler's formula provides a means of conversion between cartesian coordinates and polar coordinates. Coordinates are numbers which describe the location of points in a plane or in space The polar form reduces the number of terms from two to one, which simplifies the mathematics when used in multiplication or powers of complex numbers. The word term is from the Latin terminus "boundary line limit" from the Indo-European root ter- "peg post boundary" Any complex number z = x + iy can be written as

$z = x + iy = |z| (\cos \phi + i\sin \phi ) = |z| e^{i \phi} \,$

$\bar{z} = x - iy = |z| (\cos \phi - i\sin \phi ) = |z| e^{-i \phi} \,$

where

$x = \mathrm{Re}\{z\} \,$ the real part

$y = \mathrm{Im}\{z\} \,$ the imaginary part

$|z| = \sqrt{x^2+y^2}$ the magnitude of z

$\phi = \,$ atan2(y, x)

$\phi \,$ is the argument of z—i. The magnitude of a mathematical object is its size a property by which it can be larger or smaller than other objects of the same kind in technical terms an Ordering In Computing, atan2 is a two-argument function that makes it easy to find the angle round the origin of a point In Mathematics the arg function is a logical function that extracts the angular component of a Complex number or function e. , the angle between the x axis and the vector z measured counterclockwise and in radians—which is defined up to addition of 2π. The radian is a unit of plane Angle, equal to 180/ π degrees, or about 57 In Mathematics, the phrase " up to xxxx" indicates that members of an Equivalence class are to be regarded as a single entity for some purpose

Now, taking this derived formula, we can use Euler's formula to define the logarithm of a complex number. In Mathematics, the logarithm of a number to a given base is the power or Exponent to which the base must be raised in order to produce To do this, we also use the definition of the logarithm (as the inverse operator of exponentiation) that

$a = e^{\ln (a)}\,$

and that

$e^a e^b = e^{a + b}\,$

both valid for any complex numbers a and b.

Therefore, one can write:

$z = |z| e^{i \phi} = e^{\ln |z|} e^{i \phi} = e^{\ln |z| + i \phi}\,$

for any $z\ne 0$ . Taking the logarithm of both sides shows that:

$\ln z= \ln |z| + i \phi.\,$

and in fact this can be used as the definition for the complex logarithm. In Complex analysis, the complex logarithm is the extension of the Natural logarithm function ln( x) &ndash originally defined for Real numbers The logarithm of a complex number is thus a multi-valued function, due to the fact that $\phi \,$ is multi-valued. In Mathematics, a multivalued function (shortly multifunction, other names set-valued function, set-valued map, multi-valued map

Finally, the other exponential law

$(e^a)^k = e^{a k}, \,$

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. In Mathematics, trigonometric identities are equalities that involve Trigonometric functions that are true for every single value of the occurring variables De Moivre's formula, named after Abraham de Moivre, states that for any Complex number (and in particular for any Real number) x and any

Relationship to trigonometry

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:

$\cos x = \mathrm{Re}\{e^{ix}\} ={e^{ix} + e^{-ix} \over 2}$

$\sin x = \mathrm{Im}\{e^{ix}\} ={e^{ix} - e^{-ix} \over 2i}$

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

$e^{ix} = \cos x + i \sin x \;$

$e^{-ix} = \cos(- x) + i \sin(- x) = \cos x - i \sin x \;$

and solving for either cosine or sine. Analysis has its beginnings in the rigorous formulation of Calculus. Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. A weight function is a mathematical device used when performing a sum integral or average in order to give some elements more of a "weight" than others

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

$\cos(iy) = {e^{-y} + e^{y} \over 2} = \cosh(y)$

$\sin(iy) = {e^{-y} - e^{y} \over 2i} = i\cdot \sinh(y).$

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:

$\begin{align} \cos(x)\cdot \cos(y) & = \frac{(e^{ix}+e^{-ix})}{2} \cdot \frac{(e^{iy}+e^{-iy})}{2} \\ & = \frac{e^{i(x+y)}+e^{i(x-y)}+e^{i(-x+y)}+e^{i(-x-y)}}{4} \\ & = \frac{e^{i(x+y)}+e^{i(-x-y)}}{4}+\frac{e^{i(x-y)}+e^{i(-x+y)}}{4} \\ & = \frac{\cos(x+y)}{2} + \frac{\cos(x-y)}{2}. \end{align}$

Another technique is to represent the sinusoids in terms of the real part of a more complex expression, and perform the manipulations on the complex expression. In Mathematics, the real part of a Complex number z is the first element of the Ordered pair of Real numbers representing z For example:

$\begin{align} \cos(x\cdot n)+\cos(x\cdot(n-2)) & = \mathrm{Re} \{\quad e^{ix n}+e^{ix(n-2)}\quad \} \\ & = \mathrm{Re} \{\quad e^{ix(n-1)}\cdot (e^{ix}+e^{-ix})\quad \} \\ & = \mathrm{Re} \{\quad e^{ix(n-1)}\cdot 2\cos(x)\quad \} \\ & = \cos(x\cdot(n-1))\cdot 2\cos(x). \end{align}$

This formula is used for recursive generation of a sinusoid at intervals of x radians.

Other applications

In differential equations, the function e^ix is often used to simplify derivations, even if the final answer is a real function involving sine and cosine. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the Euler's identity is an easy consequence of Euler's formula. In Mathematical analysis, Euler's identity, named after Leonhard Euler, is the equation e^{i \pi} + 1 = 0 \\! where

In electrical engineering and other fields, signals that vary periodically over time are often described as a combination of sine and cosine functions (see Fourier analysis), and these are more conveniently expressed as the real part of exponential functions with imaginary exponents, using Euler's formula. Electrical engineering, sometimes referred to as electrical and electronic engineering, is a field of Engineering that deals with the study and application of In mathematics Fourier analysis is a subject area which grew out of the study of Fourier series Geometric interpretation Geometrically imaginary numbers are found on the vertical axis of the complex number plane Also, phasor analysis of circuits can include Euler's formula to represent the impedance of a capacitor or an inductor. In Physics and Engineering, a phase vector ("phasor" is a representation of a Sine wave whose amplitude ( A) phase ( θ)

Definitions of complex exponentiation

Main articles: Exponentiation and Exponential function

In general, raising e to a positive integer exponent has a simple interpretation in terms of repeated multiplication of e. The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) E is the fifth letter in the Latin alphabet. Its name in English is spelled e (iː plural es or ees (also written E's E In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an Raising e to zero or a negative integer exponent can be understood as repeated division. A rational number exponent can be defined by radicals of e, and an irrational number exponent can be defined by finding rational-number exponents that are arbitrarily close to the irrational-number exponent, in a limit process. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, an n th root of a Number a is a number b such that bn = a. In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction The limit of a sequence is one of the oldest concepts in Mathematical analysis. However, to define and understand a complex number exponent of e, a different type of generalization is required for the concept of exponentiation. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted

In fact, several definitions are possible. All of them can be proven to be well-defined and equivalent, although the proofs are not included in this article.

Taylor series definition

It is well-known that, for any real x, the following series is equal to e^x:

$e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$

(in other words, this is the Taylor series for the real exponential function, and it has an infinite radius of convergence). In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives In Mathematics, the radius of convergence of a Power series is a non-negative quantity either a real number or \scriptstyle \infty that represents a This invites the following definition of e^z for complex z:

$e^z = 1 + \frac{z}{1!} + \frac{z^2}{2!} + \frac{z^3}{3!} + \cdots$

This can be proven to be well-defined; in particular, the series converges for any z.

Analytic continuation definition

A simple-to-state, equivalent definition is that e^z, for complex z, is the analytic continuation of the function e^x for real x. In Complex analysis, a branch of Mathematics, analytic continuation is a technique to extend the domain of definition of a given Analytic function. This can be proven to be well-defined; in particular, it yields a single-valued function on the complex plane.

Limit definition

It is well-known that, for any real x, the following limit is equal to e^x:

$e^x = \lim_{N \rightarrow \infty} \left(1+\frac{x}{N}\right)^N$

This motivates the following definition of e^z for complex z:

$e^z = \lim_{N \rightarrow \infty} \left(1+\frac{z}{N}\right)^N$

Differential equation definition

For real x, the function f(x)=e^x is well-known to be the unique real function satisfying the differential equation:

$f'(x)=f(x),\;\;\; f(0)=1$

for all x. The limit of a sequence is one of the oldest concepts in Mathematical analysis. This motivates a definition of f(z)=e^z for complex z as the function that satisfies the differential equation:

$f'(z)=f(z),\;\;\; f(0)=1$

for all complex z, where the derivative in f'(z) is defined in the sense of a complex derivative. Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane This can be proven to yield a unique function which is well-defined everywhere on the complex plane.

Multiplicative property definition

We would expect the function e^z to have the following properties:

$e^{z_1}e^{z_2} = e^{z_1+z_2}$
$e 0 = 1$
$e 1 = e$
$e z$ is continuous

It turns out that this uniquely specifies a function on the complex plane. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output

Proofs

Various proofs of this formula are possible. The first proof below starts with the "Taylor series definition" of e^z, while the other two use the "Differential equation definition" of e^z (see above).

Using Taylor series

Here is a proof of Euler's formula using Taylor series expansions as well as basic facts about the powers of i:

$\begin{align} i^0 &{}= 1, \quad & i^1 &{}= i, \quad & i^2 &{}= -1, \quad & i^3 &{}= -i, \\ i^4 &={} 1, \quad & i^5 &={} i, \quad & i^6 &{}= -1, \quad & i^7 &{}= -i, \\ \end{align}$

and so on. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives The functions e^x, cos(x) and sin(x) (assuming x is real) can be expressed using their Taylor expansions around zero:

$\begin{align} e^x &{}= 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \\ \cos x &{}= 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \\ \sin x &{}= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \end{align}$

For complex z we define each of these functions by the above series, replacing x with z. In Mathematics, the real numbers may be described informally in several different ways This is possible because the radius of convergence of each series is infinite. In Mathematics, the radius of convergence of a Power series is a non-negative quantity either a real number or \scriptstyle \infty that represents a We then find that

$\begin{align} e^{iz} &{}= 1 + iz + \frac{(iz)^2}{2!} + \frac{(iz)^3}{3!} + \frac{(iz)^4}{4!} + \frac{(iz)^5}{5!} + \frac{(iz)^6}{6!} + \frac{(iz)^7}{7!} + \frac{(iz)^8}{8!} + \cdots \\ &{}= 1 + iz - \frac{z^2}{2!} - \frac{iz^3}{3!} + \frac{z^4}{4!} + \frac{iz^5}{5!} - \frac{z^6}{6!} - \frac{iz^7}{7!} + \frac{z^8}{8!} + \cdots \\ &{}= \left( 1 - \frac{z^2}{2!} + \frac{z^4}{4!} - \frac{z^6}{6!} + \frac{z^8}{8!} - \cdots \right) + i\left( z - \frac{z^3}{3!} + \frac{z^5}{5!} - \frac{z^7}{7!} + \cdots \right) \\ &{}= \cos (z) + i\sin (z) \end{align}$

The rearrangement of terms is justified because each series is absolutely convergent. In Mathematics, a series (or sometimes also an Integral) is said to converge absolutely if the sum (or integral of the Absolute value of the Taking z = x to be a real number gives the original identity as Euler discovered it.

Using calculus

Define the (possibly complex) function f(x), of real variable x, as

$f(x) = \frac{\cos x + i\sin x}{e^{ix}}. \$

Division by zero is precluded since the equation

$e^{ix} \cdot e^{-ix} = e^{ix \, + \, (-ix)} = e^0 = 1 \$

implies that $e^{ix} \$ is never zero.

The derivative of f(x), according to the quotient rule, is:

$\begin{align} \frac{d}{dx}f(x) &{}= \frac{e^{ix} \cdot \frac{d}{dx}(\cos x+i\sin x) - (\cos x+i\sin x) \cdot \frac{d}{dx}(e^{ix})}{(e^{ix})^2} \\ &{}= \frac{e^{ix} \cdot (-\sin x + i\cos x) - (\cos x+i\sin x) \cdot (i e^{ix})}{(e^{ix})^2} \\ &{}= \frac{-\sin x \cdot e^{ix} - i^2 \sin x \cdot e^{ix}}{(e^{ix})^2} \quad \quad \quad (i^2=-1) \\ &{}= \frac{-\sin x \cdot e^{ix} + \sin x \cdot e^{ix}}{(e^{ix})^2} \\ &{}= 0. \end{align}$

Therefore, f(x) must be a constant function in x. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Calculus, the quotient rule is a method of finding the Derivative of a function that is the Quotient of two other functions for which In Mathematics, a constant function is a function whose values do not vary and thus are Constant. Because f(0) is known, the constant that f(x) equals for all real x is also known. Thus,

$\frac{\cos x + i \sin x}{e^{ix}} = f(x) = f(0) = \frac{\cos 0 + i \sin 0}{e^0} = 1 .$

Rearranging, it follows that

$e^{ix} = \cos x + i \sin x \ .$

Q.E.D.

Using ordinary differential equations

Define the function g(x) by

$g(x) \ \stackrel{\mathrm{def}}{=}\ e^{ix} .\$

Considering that i is constant, the first and second derivatives of g(x) are

$g'(x) = i e^{ix} \$

$g''(x) = i^2 e^{ix} = -e^{ix} \$

because i ² = −1 by definition. QED is an abbreviation of the Latin phrase "la '''quod erat demonstrandum'''" which means literally "that which was to be demonstrated" From this the following 2^nd-order linear ordinary differential equation is constructed:

$g''(x) = -g(x) \$

$g''(x) + g(x) = 0. \$

Being a 2^nd-order differential equation, there are two linearly independent solutions that satisfy it:

$g_1(x) = \cos(x) \$

$g_2(x) = \sin(x). \$

Both cos(x) and sin(x) are real functions in which the 2^nd derivative is identical to the negative of that function. The word linear comes from the Latin word linearis, which means created by lines. In Mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one Independent variable, and one or more of its In Linear algebra, a family of vectors is linearly independent if none of them can be written as a Linear combination of finitely many other vectors Any linear combination of solutions to a homogeneous differential equation is also a solution. In Mathematics, linear combinations are a concept central to Linear algebra and related fields of mathematics In Mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one Independent variable, and one or more of its Then, in general, the solution to the differential equation is

$g(x)\,$	$= A g_1(x) + B g_2(x) \$
	$= A \cos(x) + B \sin(x) \$

for any constants A and B. But not all values of these two constants satisfy the known initial conditions for g(x):

$g(0) = e^{i0} = 1 \$

$g'(0) = i e^{i0} = i \$ . In Mathematics, in the field of Differential equations an initial value problem is an Ordinary differential equation together with specified value called

However these same initial conditions (applied to the general solution) are

$g(0) = A \cos(0) + B \sin(0) = A \$

$g'(0) = -A \sin(0) + B \cos(0) = B \$

resulting in

$g(0) = A = 1 \$

$g'(0) = B = i \$

and, finally,

$g(x) \ \stackrel{\mathrm{def}}{=}\ e^{ix} = \cos(x) + i \sin(x). \$