|
|
| Line 1: |
Line 1: |
| − | :''This article is about Euler's formula in ''[[complex analysis]]''. For other meanings, see [[List of topics named after Leonhard Euler|Euler function]].'' | + | Disambiguation: |
| | | | |
| − | [[Image:Euler's formula.svg|thumb|right|360px]] | + | * [[Euler's totient theorem]] |
| − | '''Euler's formula''', named after [[Leonhard Euler]], is a [[mathematics|mathematical]] formula in [[complex analysis]] that shows a deep relationship between the [[trigonometric functions]] and the [[exponential function|complex exponential function]]. ([[Euler's identity]] is a special case of the Euler formula.)
| + | * [[Euler's polyhedral formula]] |
| − | | + | * [[Euler's identity]] (<math>e^{ix}=\cos x + i\sin x</math>) |
| − | Euler's formula states that, for any [[real number]] ''x'',
| |
| − | | |
| − | : <math>e^{ix} = \cos x + i\sin x \!</math>
| |
| − | | |
| − | where
| |
| − | | |
| − | :''<math>e</math>'' is the [[e (mathematical constant)|base of the natural logarithm]]
| |
| − | | |
| − | :''<math>i</math>'' is the [[imaginary unit]]
| |
| − | | |
| − | :<math>\sin</math> and <math>\cos</math> are [[trigonometric function]]s.
| |
| − | | |
| − | [[Richard Feynman]] called Euler's formula "our jewel" and "the most remarkable formula in mathematics" (Feynman, p. 22-10).
| |
| − | | |
| − | ==History==
| |
| − | Euler's formula was [[mathematical proof|proven]] (in an obscured form) for the first time by [[Roger Cotes]] in [[1714]], then rediscovered and popularized by Euler in [[1748]]. Neither of these men saw the geometrical interpretation of the formula: the view of complex numbers as points in the plane arose only some 50 years later (see [[Caspar Wessel]]).
| |
| − | | |
| − | == Applications in complex number theory ==
| |
| − | This formula can be interpreted as saying that the function ''e''<sup>''ix''</sup> traces out the [[unit circle]] in the [[complex number|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. The formula is valid only if sin and cos take their arguments in radians rather than in degrees.
| |
| − | | |
| − | The 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''.
| |
| − | | |
| − | Euler's formula can be used to represent complex numbers in [[coordinates (elementary mathematics)|polar coordinates]]. Any complex number ''z''=''x''+''iy'' can be written as
| |
| − | | |
| − | :<math> z = x + iy = |z| (\cos \phi + i\sin \phi ) = |z| e^{i \phi} \,</math>
| |
| − | | |
| − | where
| |
| − | :<math> x = \mathrm{Re}\{z\} \,</math>
| |
| − | :<math> y = \mathrm{Im}\{z\} \,</math>
| |
| − | :<math> |z| </math> is the [[magnitude (mathematics)|magnitude]] of z
| |
| − | | |
| − | and <math>\phi</math> is the ''argument'' of ''z''— the angle between the ''x'' axis and the vector ''z'' measured counterclockwise and in [[radian]]s — which is defined [[up to]] addition of 2π.
| |
| − | | |
| − | 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
| |
| − | :<math>a = e^{ln (a)}\,</math>
| |
| − | and
| |
| − | :<math>e^a e^{b} = e^{a + b}\,</math>
| |
| − | both valid for any complex numbers ''a'' and ''b''.
| |
| − | | |
| − | Therefore, one can write:
| |
| − | :<math>
| |
| − | z=|z| e^{i \phi} =
| |
| − | e^{\ln |z|} e^{i \phi}
| |
| − | = e^{\ln |z| + i \phi}\,
| |
| − | </math>
| |
| − | | |
| − | for any <math>z\ne 0</math>. Taking the logarithm of both sides shows that:
| |
| − | : <math>\ln z= \ln |z| + i \phi.\,</math>
| |
| − | and in fact this can be used as the defintion for the [[complex logarithm]]. The logarithm of a complex number is thus a [[multi-valued function]], due to the fact that <math>\phi</math> is multi-valued.
| |
| − | | |
| − | Finally, the other exponential law
| |
| − | : <math>(e^a)^k = e^{a k}, \,</math>
| |
| − | 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]].
| |
| − | | |
| − | ==Relationship to trigonometry==
| |
| − | 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 sums of the exponential function:
| |
| − | | |
| − | : <math>\cos x = {e^{ix} + e^{-ix} \over 2} </math>
| |
| − | : <math>\sin x = {e^{ix} - e^{-ix} \over 2i}</math>
| |
| − | | |
| − | The two equations above can be derived by adding or subtracting Euler's formulas:
| |
| − | : <math>e^{ix} = \cos x + i \sin x \;</math>
| |
| − | : <math>e^{-ix} = \cos x - i \sin x \;</math>
| |
| − | 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:
| |
| − | | |
| − | :<math> \cos(iy) = {e^{-y} + e^{y} \over 2} = \cosh(y) </math>
| |
| − | | |
| − | :<math> \sin(iy) = {e^{-y} - e^{y} \over 2i} = i \sinh(y). </math>
| |
| − | | |
| − | ==Other applications==
| |
| − | 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. [[Euler's identity]] is an easy consequence of Euler's formula.
| |
| − | | |
| − | 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 number|imaginary]] exponents, using Euler's formula.
| |
| − | | |
| − | ==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'':
| |
| − | | |
| − | : <math>i^0=1 \,</math>
| |
| − | : <math>i^1=i \,</math>
| |
| − | : <math>i^2=-1 \,</math>
| |
| − | : <math>i^3=-i \,</math>
| |
| − | : <math>i^4=1 \,</math>
| |
| − | : <math>i^5=i \,</math>
| |
| − | | |
| − | and so on. The functions ''e''<sup>''x''</sup>, cos(''x'') and sin(''x'') (assuming ''x'' is [[real number|real]]) can be written as:
| |
| − | | |
| − | : <math> e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots </math>
| |
| − | | |
| − | : <math> \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots
| |
| − | </math>
| |
| − | | |
| − | : <math> \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots
| |
| − | </math>
| |
| − | | |
| − | and for complex ''z'' we ''define'' each of these function by the above series, replacing ''x'' with ''iz''. This is possible because the [[radius of convergence]] of each series is infinite. We then find that
| |
| − | | |
| − | : <math>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</math>
| |
| − | | |
| − | : <math>= 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</math>
| |
| − | | |
| − | : <math>= \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) </math>
| |
| − | | |
| − | : <math>= \cos (z) + i\sin (z) \,</math>
| |
| − | | |
| − | The rearrangement of terms is justified because each series is [[absolute convergence|absolutely convergent]]. Taking ''z'' = ''x'' to be a real number gives the original identity as Euler discovered it.
| |
| − | | |
| − | [[Q.E.D.]]
| |
| − | | |
| − | ===Using calculus===
| |
| − | Define the complex number <math>z</math> such that
| |
| − | :<math>z=\cos x + i\sin x \,</math> (ignoring the [[modulus]] term, as this cancels later)
| |
| − | Differentiating <math>z</math> with respect to <math>x</math>:
| |
| − | :<math>\frac{dz}{dx}=-\sin x + i\cos x</math>
| |
| − | Using the fact that ''i''<sup>2</sup> = -1:
| |
| − | :<math>\frac{dz}{dx}=i^2\sin x + i\cos x=i(\cos x + i\sin x)=iz</math>
| |
| − | Dividing both sides by ''z'', multiplying both sides by ''dx'', and integrating:
| |
| − | :<math>\frac{dz}{z}=idx</math>
| |
| − | :<math>\int\frac{1}{z}\,dz=\int i\,dx</math>
| |
| − | :<math>\ln z=ix + C\,</math>
| |
| − | where
| |
| − | | |
| − | :<math>C</math> is the constant of integration.
| |
| − | | |
| − | To finish the proof it must be shown that <math>C</math> is zero. This is easily done by, for example, substituting <math>x=0</math>.
| |
| − | :<math>\ln z = C\,</math>
| |
| − | But <math>z</math> is just equal to:
| |
| − | :<math>z = \cos x + i\sin x = \cos 0 + i \sin 0 = 1 \,</math>
| |
| − | thus
| |
| − | :<math>\ln 1 = C \,</math>
| |
| − | :<math>C = 0 \,</math>
| |
| − | The final step is to exponentiate
| |
| − | :<math>\ln z = ix \,</math>
| |
| − | :<math>e^{\ln z} = e^{ix} \,</math>
| |
| − | :<math>z = e^{ix} \,</math>
| |
| − | :<math>e^{ix} = \cos x + i\sin x \,</math>
| |
| − | | |
| − | ==References==
| |
| − | * Feynman, Richard P., ''The Feynman Lectures on Physics'', vol. I Addison-Wesley ([[1977]]), ISBN 0201020106, ISBN 02010211161
| |
| − | | |
| − | ==External links==
| |
| − | *[http://agutie.homestead.com/files/Eulerformula.htm Euler and his beautiful and extraordinary formula] by Antonio Gutierrez from Geometry Step by Step from the Land of the Incas.
| |
| − | *[http://agutie.homestead.com/files/Puzzle_EulerFormula.htm Euler's Formula - Puzzle: 55 pieces in a six star style of piece] by Antonio Gutierrez from Geometry Step by Step from the Land of the Incas.
| |
| − | *[http://www.DJTricities.com/eulers Detailed Proof of Euler's Relation] by Craig Lewis.
| |
| − | *[http://ccrma-www.stanford.edu/~jos/mdft/Proof_Euler_s_Identity.html Proof of Euler's Formula] by Julius O. Smith III
| |
| − | *[http://fermatslasttheorem.blogspot.com/2006/02/eulers-formula.html Euler's Formula and Fermat's Last Theorem]
| |
| − | | |
| − | ==See also==
| |
| − | * [[Leonhard Euler]]
| |
| − | * [[Euler's identity]]
| |
| − | * [[Complex number]]
| |
| − | * [[Exponential function]]
| |
| − | * [[Trigonometry]]
| |
| − | | |
| − | [[Category:Complex analysis]]
| |
| − | [[Category:Mathematical theorems]]
| |
)
