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| :''This article is about Euler's formula in complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see [[Euler characteristic]].
| | If you've ever tried to get rid of weight then we know that low calorie diets may only receive you thus far. Eventually, your body will grow accustomed to the lower level of calories plus a metabolism can slow down accordingly. That's when you reach a fat reduction plateau and it can seem because when regardless what you do, the fat won't come off. This is a ideal time to test a carb cycling plan for weight reduction. Carb cycling plans have been selected by body builders plus fitness models for some time now however haven't caught on quite promptly among the general public. It's a shame considering carb cycling for weight reduction works extremely perfectly! If you follow this type of program, you'll essentially trick your body into burning fat, enabling we to reach the weight loss goals. Give it a try for quick weight reduction.<br><br>There are steps that we must do so that you will lose body fat. You can become successful inside calorie burn calculator achieving a goals when you may be determine plus committed to do all procedures.<br><br>Calorie Counter (click here) is a free online weight loss resource website. Calorie Counter offers free calorie charts, nutrition information banks, diet plus food journals and additional diet support tools. User must register and log in to access full website advantages, nevertheless there is not any cost. The webpage features an an interactive movie to explain how to employ the food log tool called "Food Logger". There is furthermore a comprehensive list of foods, menu items, brand names, diners foods plus treatments which dieters could use to check calorie count and nutrition information. This website has free online and printable food journals, exercise plans, workout schedule, fitness suggestions, weight reduction recipes plus strategies and more.<br><br>Considering that most dieters detest [http://safedietplansforwomen.com/calories-burned-walking calorie burn calculator] to do anything that is not fun, they consistently avoid jogging. And they miss the chance to burn calories.<br><br>Whenever the body senses which it is being starved, it goes into "starvation mode". To safeguard itself against starvation again, the body may automatically store much of the calories burned calculator last meal because a fat reserve to draw from later. You've only told your body to store fat for the next time we plan on starving it of important vitamins.<br><br>Remember, no body is the same, thus there is no exact means to count calories, or to plug in to several calorie calculator. This formula is a standard guideline that may create eating less or perhaps a secret and more of the necessity with allowances plus healthy outcomes.<br><br>But regardless of which activity we eventually choose, running or strolling, the significant thing to remember is the fact that leading an active life is important to building an total state of wellness. The key to staying healthy is to keep moving. |
| {{E (mathematical constant)}}
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| '''Euler's formula''', named after [[Leonhard Euler]], is a [[mathematics|mathematical]] [[formula]] in [[complex analysis]] that establishes the fundamental relationship between the [[trigonometric functions]] and the [[complex number|complex]] [[exponential function]]. Euler's formula states that, for any [[real number]] ''x'',
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| : <math>e^{ix} = \cos x + i\sin x \ </math>
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| where ''e'' is the [[e (mathematical constant)|base of the natural logarithm]], ''i'' is the [[imaginary unit]], and cos and sin are the [[trigonometric functions]] cosine and sine respectively, with the argument ''x'' given in [[radian]]s. This complex exponential function is sometimes denoted {{nobreak|'''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.<ref>{{cite book | first=Martin A. | last= Moskowitz | title=A Course in Complex Analysis in One Variable | publisher=World Scientific Publishing Co. | year=2002 | isbn=981-02-4780-X | pages=7}}</ref>
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| 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."<ref>{{cite book|first=Richard P.|last= Feynman|title=The Feynman Lectures on Physics, vol. I|publisher=Addison-Wesley|year=1977|isbn=0-201-02010-6|page=22-10}}</ref>
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| ==History==
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| It was [[Johann Bernoulli]] who noted that<ref>Johann Bernoulli, Solution d'un problème concernant le calcul intégral, avec quelques abrégés par rapport à ce calcul, ''Mémoires de l'Académie Royale des Sciences de Paris'', 197-289 (1702).</ref>
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| :<math>\frac{1}{1+x^2}=\frac{1}{2} \left(\frac{1}{1-ix}+\frac{1}{1+ix} \right) \ .</math>
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| And since
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| : <math>\int \frac{dx}{1+ax}=\frac{1}{a}\ln(1+ax)+C \ ,</math>
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| the above equation tells us something about [[complex logarithm]]s. Bernoulli, however, did not evaluate the integral.
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| Bernoulli's correspondence with Euler (who also knew the above equation) shows that Bernoulli did not fully understand [[complex logarithm]]s. Euler also suggested that the complex logarithms can have infinitely many values.
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| Meanwhile, [[Roger Cotes]], in 1714, discovered that
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| :<math> \ln(\cos x + i\sin x)=ix \ </math>
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| ("ln" is the [[natural logarithm]] with base ''e'').<ref name="Stillwell">{{cite book|author=John Stillwell|title=Mathematics and Its History|publisher=Springer|year=2002 | url = http://books.google.com/books?id=V7mxZqjs5yUC&pg=PA315}}</ref>
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| Cotes missed the fact that a complex logarithm can have infinitely many values, differing by multiples of 2{{pi}}, due to the periodicity of the trigonometric functions.
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| Around 1740 Euler turned his attention to the exponential function instead of logarithms, and obtained the formula used today that is named after him. It was published in 1748, obtained by comparing the series expansions of the exponential and trigonometric expressions.<ref name="Stillwell"/>
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| None of these mathematicians saw the geometrical interpretation of the formula; the view of complex numbers as points in the [[complex plane]] was described some 50 years later by [[Caspar Wessel]].
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| ==Applications in complex number theory==
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| [[Image:Euler's formula.svg|thumb
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| |right]]
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| [[Image:Euler's Formula c.png|thumb|upright=1.5|Three-dimensional visualization of Euler's formula. See also [[circular polarization]].]]
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| This formula can be interpreted as saying that the function ''e''<sup>''ix''</sup> 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 [[radian]]s. | |
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| The original proof is based on the [[Taylor series]] expansions of the [[exponential function]] ''e''<sup>''z''</sup> (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''.
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| A point in the [[complex plane]] can be represented by a complex number written in
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| [[Coordinates (elementary_mathematics)#Cartesian coordinates|cartesian coordinates]]. Euler's formula provides a means of conversion between cartesian coordinates and [[coordinates (elementary mathematics)#Polar coordinates|polar coordinates]]. The polar form simplifies the mathematics when used in multiplication or powers of complex numbers. Any complex number ''z'' = ''x'' + ''iy'' can be written as
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| :<math> z = x + iy = |z| (\cos \phi + i\sin \phi ) = r e^{i \phi} \ </math>
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| :<math> \bar{z} = x - iy = |z| (\cos \phi - i\sin \phi ) = r e^{-i \phi} \ </math>
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| where
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| :<math> x = \mathrm{Re}\{z\} \,</math> the real part
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| :<math> y = \mathrm{Im}\{z\} \,</math> the imaginary part
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| :<math> r = |z| = \sqrt{x^2+y^2}</math> the [[magnitude (mathematics)|magnitude]] of ''z''
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| :<math>\phi = \arg z = \,</math> [[atan2]](''y'', ''x'') .
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| <math>\phi \,</math> is the ''[[arg (mathematics)|argument]]'' of ''z''—i.e., the angle between the ''x'' axis and the vector ''z'' measured counterclockwise and in [[radian]]s—which is defined [[up to]] addition of 2π. Many texts write tan<sup>−1</sup>(''y''/''x'') instead of atan2(''y'',''x'') but this needs adjustment when ''x'' ≤ 0.
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| 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) that
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| :<math>a = e^{\ln (a)} \ </math>
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| and that
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| :<math>e^a e^b = e^{a + b} \ </math>
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| both valid for any complex numbers ''a'' and ''b''.
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| Therefore, one can write:
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| :<math> z = |z| e^{i \phi} = e^{\ln |z|} e^{i \phi} = e^{\ln |z| + i \phi} \ </math>
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| for any ''z'' ≠ 0. Taking the logarithm of both sides shows that:
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| : <math>\ln z= \ln |z| + i \phi \ .</math>
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| 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 <math>\phi</math> is multi-valued.
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| Finally, the other exponential law
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| : <math>(e^a)^k = e^{a k} \ ,</math>
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| which can be seen to hold for all integers ''k'', together with Euler's formula, implies several [[trigonometric identity|trigonometric identities]] as well as [[de Moivre's formula]].
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| ==Relationship to trigonometry==
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| [[File:Sine Cosine Exponential qtl1.svg|thumb|Relationship between sine, cosine and exponential function]]
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| Euler's formula provides a powerful connection between [[mathematical analysis|analysis]] and [[trigonometry]], and provides an interpretation of the sine and cosine functions as [[weighted sum]]s of the exponential function''':'''
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| : <math>\cos x = \mathrm{Re}\{e^{ix}\} ={e^{ix} + e^{-ix} \over 2}</math>
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| : <math>\sin x = \mathrm{Im}\{e^{ix}\} ={e^{ix} - e^{-ix} \over 2i} </math>
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| The two equations above can be derived by adding or subtracting Euler's formulas''':'''
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| : <math>e^{ix} = \cos x + i \sin x \;</math>
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| : <math>e^{-ix} = \cos(- x) + i \sin(- x) = \cos x - i \sin x \;</math>
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| and solving for either cosine or sine.
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| These formulas can even serve as the definition of the trigonometric functions for complex arguments ''x''. For example, letting ''y'' = ''ix'', we have''':'''
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| :<math> \cos(iy) = {e^{-y} + e^{y} \over 2} = \cosh(y) </math>
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| :<math> \sin(iy) = {e^{-y} - e^{y} \over 2i} = - {e^{y} - e^{-y} \over 2i} = i\sinh(y) \ . </math>
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| 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''':'''
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| : <math>
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| \begin{align}
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| \cos x\cdot \cos y & = \frac{(e^{ix}+e^{-ix})}{2} \cdot \frac{(e^{iy}+e^{-iy})}{2} \\
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| & = \frac{1}{2}\cdot \frac{e^{i(x+y)}+e^{i(x-y)}+e^{i(-x+y)}+e^{i(-x-y)}}{2} \\
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| & = \frac{1}{2} \bigg[ \underbrace{ \frac{e^{i(x+y)} + e^{-i(x+y)}}{2} }_{\cos(x+y)} + \underbrace{ \frac{e^{i(x-y)} + e^{-i(x-y)}}{2} }_{\cos(x-y)} \bigg] \
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| \end{align}
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| </math>
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| 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''':'''
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| : <math>
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| \begin{align}
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| \cos(nx) & = \mathrm{Re} \{\ e^{inx}\ \}
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| = \mathrm{Re} \{\ e^{i(n-1)x}\cdot e^{ix}\ \} \\
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| & = \mathrm{Re} \{\ e^{i(n-1)x}\cdot (\underbrace{e^{ix} + e^{-ix}}_{2\cos(x)} - e^{-ix})\ \} \\
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| & = \mathrm{Re} \{\ e^{i(n-1)x}\cdot 2\cos(x) - e^{i(n-2)x}\ \} \\
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| & = \cos[(n-1)x]\cdot 2 \cos(x) - \cos[(n-2)x] \
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| \end{align}</math>
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| This formula is used for recursive generation of cos(''nx'') for integer values of ''n'' and arbitrary ''x'' (in radians).
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| See also [[Phasor#Phasor_arithmetic|Phasor arithmetic]].
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| ==Topological interpretation==
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| In the language of [[topology]], Euler's formula states that the imaginary exponential function <math>t\mapsto e^{it}</math> is a (surjective) [[morphism]] of [[Topological group|topological groups]] from the real line <math>\mathbb R</math> to the unit circle <math>\mathbb S^1</math>. In fact, this exhibits <math>\mathbb R</math> as a [[covering space]] of <math>\mathbb S^1</math>. Similarly, [[Euler's identity]] says that the [[Kernel (algebra)|kernel]] of this map is <math>\tau\mathbb Z</math>, where <math>\tau = 2\pi</math>. These observations may be combined and summarized in the [[commutative diagram]] below:
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| [[File:Euler's formula commutative diagram.png|frameless|center|Euler's formula and identity combined in diagrammatic form]]
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| ==Other applications==
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| In [[differential equations]], the function ''e''<sup>''ix''</sup> is often used to simplify derivations, even if the final answer is a real function involving sine and cosine. The reason for this is that the complex exponential is the [[eigenfunction]] of differentiation. [[Euler's identity]] is an easy consequence of Euler's formula.
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| In [[electronic 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 number|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.
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| ==Definitions of complex exponentiation==
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| {{main|Exponentiation|Exponential function}}
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| The exponential function ''e<sup>x</sup>'' 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<sup>z</sup>'' for complex values of ''z'' simply by substituting ''z'' in place of ''x'' and using the complex algebraic operations. In particular we may use either of the two following definitions which are equivalent. From a more advanced perspective, each of these definitions may be interpreted as giving the unique [[analytic continuation]] of ''e<sup>x</sup>'' to the complex plane.
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| ===Power series definition===
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| For complex ''z''
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| :<math>e^z = 1 + \frac{z}{1!} + \frac{z^2}{2!} + \frac{z^3}{3!} + \dots = \sum_{n=0}^{\infty} \frac{z^n}{n!}.</math>
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| Using the [[ratio test]] it is possible to show that this [[power series]] has an infinite [[radius of convergence]], and so defines ''e<sup>z</sup>'' for all complex ''z''.
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| ===Limit definition===
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| For complex ''z''
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| :<math>e^z = \lim_{n \rightarrow \infty} \left(1+\frac{z}{n}\right)^n ~.</math>
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| ==Proofs==
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| Various proofs of the formula are possible.
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| ===Using power series===
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| Here is a proof of Euler's formula using [[Taylor series|power series expansions]]
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| as well as basic facts about the powers of ''i'':<ref>[http://books.google.com/books?id=PjK0F0T3NBoC&pg=PA428 A Modern Introduction to Differential Equations, by Henry J. Ricardo, p428]</ref>
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| : <math>\begin{align}
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| i^0 &{}= 1, \quad &
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| i^1 &{}= i, \quad &
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| i^2 &{}= -1, \quad &
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| i^3 &{}= -i, \\
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| i^4 &={} 1, \quad &
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| i^5 &={} i, \quad &
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| i^6 &{}= -1, \quad &
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| i^7 &{}= -i,
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| \end{align}</math>
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| and so on. Using now the power series definition from above we see that for real values of ''x''
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| : <math>\begin{align}
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| e^{ix} &{}= 1 + ix + \frac{(ix)^2}{2!} + \frac{(ix)^3}{3!} + \frac{(ix)^4}{4!} + \frac{(ix)^5}{5!} + \frac{(ix)^6}{6!} + \frac{(ix)^7}{7!} + \frac{(ix)^8}{8!} + \cdots \\[8pt]
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| &{}= 1 + ix - \frac{x^2}{2!} - \frac{ix^3}{3!} + \frac{x^4}{4!} + \frac{ix^5}{5!} - \frac{x^6}{6!} - \frac{ix^7}{7!} + \frac{x^8}{8!} + \cdots \\[8pt]
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| &{}= \left( 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!} - \cdots \right) + i\left( x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \right) \\[8pt]
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| &{}= \cos x + i\sin x \ .
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| \end{align}</math>
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| In the last step we have simply recognized the [[Taylor series|Maclaurin series]] for ''cos(x)'' and ''sin(x)''. The rearrangement of terms is justified because each series is [[absolute convergence|absolutely convergent]].
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| ===Using the limit definition===
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| [[File:EulerFormulaAsLimit.gif|thumb|right|The [[exponential function]] {{math|''e''}}<sup>{{math|''z''}}</sup> can be defined as the [[limit of a sequence|limit]] of {{math|(1 + ''z''/''n'')<sup>''n''</sup>}}, as {{math|''n''}} approaches infinity. In this animation, {{math|''z''{{=}}''iπ''/3</sup>}}, and {{math|''n''}} takes various increasing values from 1 to 100. The computation of {{math|(1 + ''z''/''n'')<sup>''n''</sup>}} is displayed as the combined effect of {{math|''n''}} repeated multiplications in the [[complex plane]]. As {{math|''n''}} gets larger, the points approach the complex [[unit circle]] (dashed line), covering an angle of {{math|''π''/3}} radians.]]
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| An alternative proof<ref name=Arnold>[http://books.google.com/books?id=JUoyqlW7PZgC&pg=PA166 ''Ordinary differential equations'', by Vladimir Igorevich Arnolʹd, p166]</ref> starts from the limit definition of <math>e^z</math>:
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| :<math>e^z = \lim_{n\rightarrow\infty} \left(1+\frac{z}{n}\right)^n</math>.
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| Substitute <math>z=ix</math>, and let {{math|''n''}} be a very large integer. Then consider the sequence:
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| :<math> 1, \, \left(1+\frac{ix}{1}\right)^1, \, \left(1+\frac{ix}{2}\right)^2, \ldots, \, \left(1+\frac{ix}{n}\right)^n </math>
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| (The last element of the sequence approaches {{math|''e''<sup>''ix''</sup>}}.) If the points of this sequence are plotted in the [[complex plane]] (see animation at right), they roughly trace out the [[unit circle]], with each point being {{math|''x''/''n''}} radians counterclockwise of the previous point. (This statement is more and more accurate as {{math|''n''}} increases. The proof is based on the rules of trigonometry and complex-number algebra.<ref name=Arnold/>) Therefore, in the limit {{math|''n''→∞}}, the last point in the sequence, {{math|(1 + ''ix''/''n'')<sup>''n''</sup>}}, is the point on the [[unit circle]] of the [[complex plane]] located {{math|''x''}} radians counterclockwise from +1, that is the point {{math|cos ''x'' + ''i'' sin ''x''}}. Therefore, {{math|1=''e''<sup>''ix''</sup> = cos ''x'' + ''i'' sin ''x''}}.
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| ===Using calculus===
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| Another proof<ref name=Strang>{{cite book |url=http://ocw.mit.edu/resources/res-18-001-calculus-online-textbook-spring-2005/textbook/ |title=Calculus |first=Gilbert |last=Strang |page=389 |publisher=Wellesley-Cambridge |year=1991 |isbn=0-9614088-2-0}} ''(Second proof on page)''</ref> is based on the fact that all complex numbers can be expressed in polar coordinates. Therefore for some <math>r</math> and <math>\theta</math> depending on <math>x</math>,
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| :<math>e^{ix} = r (\cos(\theta) + i \sin(\theta))\,.</math>
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| Now from any of the definitions of the exponential function it can be shown that the derivative of <math>e ^{ix}</math> is <math>i e ^{ix}</math>. Therefore differentiating both sides gives
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| :<math>i e ^{ix} = (\cos(\theta) + i \sin(\theta)) \frac{dr}{dx} + r (-\sin(\theta) + i \cos(\theta)) \frac{d \theta}{dx}\,.</math>
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| Substituting <math>r (\cos(\theta) + i \sin(\theta))</math> for <math>e^{ix}</math> and equating real and imaginary parts in this formula gives <math>\textstyle \frac{dr}{dx} = 0</math> and <math>\textstyle \frac{d\theta}{dx} = 1</math>. Together with the initial values <math>r(0) = 1</math> and <math>\theta(0) = 0</math> which come from <math>e^{i0} = 1</math> this gives <math>r=1</math> and <math>\theta=x</math>. This proves the formula <math>e^{ix} = 1(\cos(x)+i \sin(x))</math>.
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| ==See also==
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| * [[Complex number]]
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| * [[Euler's identity]]
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| * [[Integration using Euler's formula]]
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| * [[List of topics named after Leonhard Euler]]
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| ==References==
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| <references/>
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| ==External links==
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| *{{springer|title=Euler formulas|id=p/e036460}}
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| *[http://ccrma-www.stanford.edu/~jos/mdft/Proof_Euler_s_Identity.html Proof of Euler's Formula] by Julius O. Smith III
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| *[http://fermatslasttheorem.blogspot.com/2006/02/eulers-formula.html Euler's Formula and Fermat's Last Theorem]
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| *[http://math.fullerton.edu/mathews/c2003/ComplexFunExponentialMod.html Complex Exponential Function Module by John H. Mathews]
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| *[http://web.mat.bham.ac.uk/C.J.Sangwin/euler/ Elements of Algebra]
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| *[http://resonanceswavesandfields.blogspot.com/2007/08/eulers-equation-and-complex-numbers.html Visual Derivation of Euler's Formula]
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| *[http://www.khanacademy.org/math/calculus/v/euler-s-formula-and-euler-s-identity Euler's Formula and Euler's Identity : Rationale for Euler's Formula and Euler's Identity, video at Khanacademy.org]
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| {{Use dmy dates|date=August 2011}}
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| *[http://www.docstoc.com/docs/159557614/Exponential-Circular-Integrals/ Difficult definite integrals by complex numbers]
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| [[Category:Complex analysis]]
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| [[Category:Theorems in complex analysis]]
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| [[Category:Articles containing proofs]]
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| [[Category:E (mathematical constant)]]
| |
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