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Encyclopedia > Euler's formula

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. (Euler's identity is a special case of the Euler formula.) Image File history File links Euler's_formula. ... Image File history File links Euler's_formula. ... Leonhard Paul Euler (pronounced Oiler; IPA ) (April 15, 1707 – September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics and in the sciences, a formula (plural: formulae, formulæ or formulas) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. ... Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... The exponential function is one of the most important functions in mathematics. ... For other meanings, see List of topics named after Leonhard Euler In mathematical analysis, Eulers identity, named after Leonhard Euler, is the equation where is Eulers number, the base of the natural logarithm, is the imaginary unit, one of the two complex numbers whose square is negative one...

  Part of a series of articles on
The mathematical constant, e e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ...

Natural logarithm Image File history File links Euler's_formula. ... The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is equal to 2. ...

Applications in Compound interest · Euler's identity & Euler's formula  · Half lives & Exponential growth/decay Compound interest refers to the fact that whenever interest is calculated, it is based not only on the original principal, but also on any unpaid interest that has been added to the principal. ... For other meanings, see List of topics named after Leonhard Euler In mathematical analysis, Eulers identity, named after Leonhard Euler, is the equation where is Eulers number, the base of the natural logarithm, is the imaginary unit, one of the two complex numbers whose square is negative one... Half-Life For a quantity subject to exponential decay, the half-life is the time required for the quantity to fall to half of its initial value. ... In mathematics, exponential growth (or geometric growth) occurs when the growth rate of a function is always proportional to the functions current size. ... A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. ...

Defining e Proof that e is irrational  · Representations of e · Lindemann–Weierstrass theorem In mathematics, the series expansion of the number e can be used to prove that e is irrational. ... The mathematical constant e can be represented in a variety of ways as a real number. ... In mathematics, the Lindemann–Weierstrass theorem states that if α1,...,αn are algebraic numbers which are linearly independent over the rational numbers, then are algebraically independent over the algebraic numbers; in other words the set has transcendence degree n over . ...

People John Napier  · Leonhard Euler
For other people with the same name, see John Napier (disambiguation). ... Leonhard Paul Euler (pronounced Oiler; IPA ) (April 15, 1707 – September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ...

Schanuel's conjecture
Schanuels conjecture is that given any set of n complex numbers which have linear independence over the rational numbers, the set (up to twice the size) has transcendence degree of at least n over the rationals. ...

Euler's formula states that, for any real number x, In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

e^{ix} = cos(x) + isin(x) !

where

 e , is the base of the natural logarithm
 i , is the imaginary unit
 mathrm{cos} , and  mathrm{sin} , are trigonometric functions.

Richard Feynman called Euler's formula "our jewel" and "the most remarkable formula in mathematics".[1] e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ... In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . ... In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. ... Richard Phillips Feynman (May 11, 1918 – February 15, 1988; IPA: ) was an American physicist known for expanding the theory of quantum electrodynamics, the physics of the superfluidity of supercooled liquid helium, and particle theory. ...

Contents

History

Euler's formula was proven for the first time by Roger Cotes in 1714 in the form In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ... Roger Cotes (Burbage, Leicestershire July 10, 1682 – June 5, 1716 in Cambridge, Cambridgeshire) was an English mathematician. ... Battle of Gangut, by Maurice Baquoi, 1724-27. ...

 ln(cos(x) + isin(x))=ix

(where "ln" means natural logarithm, i.e. log with base e)[2]. The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is equal to 2. ...


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. 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). 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. Events April 24 - A congress assembles at Aix-la-Chapelle with the intent to conclude the struggle known as the War of Austrian Succession - at October 18 - The Treaty of Aix-la-Chapelle is signed to end the war Adam Smith begins to deliver public lectures in Edinburgh Building of... In mathematics, a series is a sum of a sequence of terms. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... Caspar Wessel (June 8, 1745 - March 25, 1818) was a Norwegian-Danish mathematician. ...


Applications in complex number theory

This formula can be interpreted as saying that the function eix traces out the unit circle in the complex number plane as x ranges through the real numbers. 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. The formula is valid only if sin and cos take their arguments in radians rather than in degrees. Illustration of a unit circle. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... ∠, the angle symbol. ... Some common angles, measured 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 z. As the degree of the Taylor series rises, it approaches the correct function. ... The exponential function is one of the most important functions in mathematics. ...


Euler's formula can be used to represent complex numbers in polar coordinates. Any complex number z = x + iy can be written as This article describes some of the common coordinate systems that appear in elementary mathematics. ...

 z = x + iy = |z| (cos phi + isin phi ) = |z| e^{i phi} ,
 bar{z} = x - iy = |z| (cos phi - isin 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

and  phi , is the argument of z— the angle between the x axis and the vector z measured counterclockwise and in radians — which is defined up to addition of 2π. 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 of the class of objects to which it belongs. ... Some common angles, measured in radians. ... Look up Up to on Wiktionary, the free dictionary 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. To do this, we also use the facts that Logarithms to various bases: is to base e, is to base 10, and is to base 1. ...

a = e^{ln (a)},

and

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 zne 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. The logarithm of a complex number is thus a multi-valued function, due to the fact that phi , is multi-valued. The natural logarithm is the logarithm to the base e, where e is equal to 2. ... This diagram does not represent a true function; because the element 3, in X, is associated with two elements b and c, in Y. In mathematics, a multivalued function is a total relation; i. ...


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 equations involving trigonometric functions that are true for all values of the occurring variables. ... de Moivres formula, named after Abraham de Moivre, states that for any complex number (and, in particular, for any real number) x and any integer n it holds that The formula is important because it connects complex numbers (i stands for the imaginary unit) and trigonometry. ...


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: Analysis has its beginnings in the rigorous formulation of calculus. ... Wikibooks has a book on the topic of Trigonometry All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Trigonometry (from Greek trigōnon triangle + metron measure[1]), informally called trig, is a branch of mathematics that deals with... 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. ...

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.


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 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. For example: In mathematics, the real part of a complex number , is the first element of the ordered pair of real numbers representing , i. ...

 begin{align} cos(xcdot n)+cos(x(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 2cos(x)quad }  & = cos(x(n-1))cdot 2cos(x). end{align}

Other applications

In differential equations, the function eix is often used to simplify derivations, even if the final answer is a real function involving sine and cosine. Euler's identity is an easy consequence of Euler's formula. In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... For other meanings, see List of topics named after Leonhard Euler In mathematical analysis, Eulers identity, named after Leonhard Euler, is the equation where is Eulers number, the base of the natural logarithm, is the imaginary unit, one of the two complex numbers whose square is negative one...


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. Also, phasor analysis of circuits can include Euler's formula to represent the impedance of a capacitor or an inductor. Electrical Engineers design power systems… … and complex electronic circuits. ... Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ... It has been suggested that this article or section be merged with Phasor (physics). ...


Proofs

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: As the degree of the Taylor series rises, it approaches the correct function. ...

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. The functions ex, cos(x) and sin(x) (assuming x is real) can be expressed using their Taylor expansions around zero: In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

 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 function by the above series, replacing x with z. This is possible because the radius of convergence of each series is infinite. We then find that In mathematics, the radius of convergence of a power series where the center a and the coefficients cn are complex numbers (which may, in particular, be real numbers) is the nonnegative quantity r (which may be a real number or ∞) such that the series converges if and diverges if In...

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) + ileft( z - frac{z^3}{3!} + frac{z^5}{5!} - frac{z^7}{7!} + cdots right)  &{}= cos (z) + isin (z) end{align}

The rearrangement of terms is justified because each series is absolutely convergent. Taking z = x to be a real number gives the original identity as Euler discovered it. In mathematics, a series is a sum of a sequence of terms. ...


Using calculus

Define the function f by

f(x) = frac{cos x+isin x}{e^{ix}}.

This is allowed since the equation

e^{ix}cdot e^{-ix}=e^0=1

implies that eix is never zero.


The derivative of f, according to the quotient rule, is: For a non-technical overview of the subject, see Calculus. ... 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 derivatives exist. ...

begin{align} f'(x) &{}= frac{(-sin x+icos x)cdot e^{ix} - (cos x+isin x)cdot icdot e^{ix}}{(e^{ix})^2}  &{}= frac{-sin xcdot e^{ix}-i^2sin xcdot e^{ix}}{(e^{ix})^2}  &{}= frac{(-1 - i^2) cdot sin x cdot e^{ix}}{(e^{ix})^2}  &{}= frac{(-1 - (-1)) cdot sin x cdot e^{ix}}{(e^{ix})^2}  &{}= 0. end{align}

Therefore, f  must be a constant function. Thus, In mathematics a constant function is a function whose values do not vary and thus are constant. ...

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

displaystylecos x + i sin x=e^{ix} .

Q.E.D. Look up QED in Wiktionary, the free dictionary. ...


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. From this the following 2nd-order linear ordinary differential equation is constructed: 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 derivatives with respect to that variable. ...

g''(x) = -g(x)

or

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

Being a 2nd-order differential equation, there are two linearly independent solutions that satisfy it: In linear algebra, a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set. ...

g_1(x) = cos(x)
g_2(x) = sin(x).

Both cos(x) and sin(x) are real functions in which the 2nd derivative is identical to the negative of that function. Any linear combination of solutions to a homogeneous differential equation is also a solution. Then, in general, the solution to the differential equation is 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 derivatives with respect to that variable. ...

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): In mathematics, boundary conditions are imposed on the solutions of ordinary differential equations and partial differential equations, to fit the solutions to the actual problem. ...

g(0) = e^{i0} = 1
g'(0) = i e^{i0} = i  .

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).

Q.E.D. Look up QED in Wiktionary, the free dictionary. ...


See also

Leonhard Paul Euler (pronounced Oiler; IPA ) (April 15, 1707 – September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... For other meanings, see List of topics named after Leonhard Euler In mathematical analysis, Eulers identity, named after Leonhard Euler, is the equation where is Eulers number, the base of the natural logarithm, is the imaginary unit, one of the two complex numbers whose square is negative one... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... Exponentiation is a mathematical operation, written an, involving two numbers, the base a and the exponent n. ... The exponential function is one of the most important functions in mathematics. ... Wikibooks has a book on the topic of Trigonometry All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Trigonometry (from Greek trigōnon triangle + metron measure[1]), informally called trig, is a branch of mathematics that deals with...

References

  1. ^ Feynman, Richard P. (1977). The Feynman Lectures on Physics, vol. I. Addison-Wesley, p. 22-10. ISBN 0-201-02010-6. 
  2. ^ John Stillwell (2002). Mathematics and Its History. Springer. 

External links


  Results from FactBites:
 
Euler's formula - Wikipedia, the free encyclopedia (846 words)
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.
Euler's formula was proven (in an obscured form) for the first time by Roger Cotes in 1714, then rediscovered and popularized by Euler in 1748.
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:
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