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Difference between revisions of "Euler's identity"

 
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Euler's formula is <math>\displaystyle e^{i\theta}=\cos(\theta)+i\sin(\theta)</math>. This can be shown using [[infinite power series]] for <math>e^x, \sin(x)</math>, and <math>\cos(x)</math>.
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'''Euler's Formula''' is <math>e^{i\theta}=\cos \theta+ i\sin\theta</math>.   It is named after the 18th-century mathematician [[Leonhard Euler]].
  
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== Proof ==
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==Background==
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Note that
 
  
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<math>e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...=\sum_{k=0}^{\infty}\frac{x^n}{n!}</math>
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Euler's formula is a fundamental tool used when solving problems involving [[complex numbers]] and/or [[trigonometry]]. Euler's formula replaces "[[cis]]", and is a superior notation, as it encapsulates several nice properties:
  
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<math>\sin(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-...=\sum_{j=0}^{\infty}(-1)^{j}\frac{x^{2j+1}}{(2j+1)!}</math>
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===De Moivre's Theorem===
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[[De Moivre's Theorem]] states that for any [[real number]] <math>\theta</math> and integer <math>n</math>,
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<math>(\cos(\theta) + i\sin(\theta))^n = (e^{i\theta})^n = e^{in\theta} = \cos(n\theta) + i\sin(n\theta)</math>.
  
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<math>\cos(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-...=\sum_{i=0}^{\infty}(-1)^{i}\frac{x^{2n}}{(2n)!}</math>
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===Sine/Cosine Angle Addition Formulas===
  
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(where i, j, k are just [[dummy variables]]).
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Start with <math>e^{i(\alpha + \beta)} = (e^{i\alpha})(e^{i\beta})</math>, and apply Euler's forumla both sides:
  
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The key step now is to let <math>x=i\theta</math> and plug it into the series for <math>e^x</math>.  The result is Euler's formula above. (anyone who's willing, feel free to type up the steps).
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<math>
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\cos(\alpha + \beta) + i \sin(\alpha + \beta) = (\cos\alpha + i\sin\alpha)(\cos\beta + i\sin\beta).</math>
  
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A special, and quite fascinating, consequence of Euler's formula is the identity <math>e^{i\pi}+1=0</math>, which relates five of the most fundamental numbers in all of mathematics: <math>e,i,\pi, 0</math> and 1.
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Expanding the right side gives
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<math>
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(\cos\alpha\cos\beta - \sin\alpha\sin\beta) + i(\cos\alpha\sin\beta + \sin\alpha\cos\beta).
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</math>
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Comparing the real and imaginary terms of these expressions gives the sine and cosine angle-addition formulas:
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<math>
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\cos(\alpha+\beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta
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</math>
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<math>
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\sin(\alpha+\beta) = \cos\alpha\sin\beta + \sin\alpha\cos\beta
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</math>
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===Geometry on the complex plane===
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===Other nice properties===
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A special, and quite fascinating, consequence of Euler's formula is the identity <math>e^{i\pi}+1=0</math>, which relates five of the most fundamental numbers in all of mathematics: [[e]], [[imaginary unit | i]], [[pi]], [[zero (constant)| 0]], and 1.
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==Proof 1==
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The proof of Euler's formula can be shown using the technique from [[calculus]] known as [[Taylor series]].
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We have the following Taylor series:
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<math>e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots=\sum_{k=0}^{\infty}\frac{x^k}{k!}</math>
 +
 
 +
<math>\sin(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots=\sum_{k=0}^{\infty}(-1)^{k}\frac{x^{2k+1}}{(2k+1)!}</math>
 +
 
 +
<math>\cos(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots=\sum_{k=0}^{\infty}(-1)^{k}\frac{x^{2k}}{(2k)!}</math>
 +
 
 +
The key step now is to let <math>x=i\theta</math> and plug it into the series for <math>e^x</math>.  The result is Euler's formula above.
 +
 
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==Proof 2==
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Define <math>z=\cos{\theta}+i\sin{\theta}</math>. Then <math>\frac{dz}{d\theta}=-\sin{\theta}+i\cos{\theta}=iz</math>, <math>\implies \frac{dz}{z}=id\theta</math>
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<math>\int \frac{dz}{z}=\int id\theta</math>
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<math>\ln{|z|}=i\theta+c</math>
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<math>z=e^{i\theta+c}</math>; we know <math>z(0)=1</math>, so we get <math>c=0</math>, therefore <math>z=e^{i\theta}=\cos{\theta}+i\sin{\theta}</math>.
  
 
== See Also ==
 
== See Also ==
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*[[Common power series]]
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*[[Complex numbers]]
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*[[Trigonometry]]
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*[[Power series]]
 
*[[Convergence]]
 
*[[Convergence]]
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[[Category:Complex numbers]]

Latest revision as of 22:17, 4 January 2021

Euler's Formula is $e^{i\theta}=\cos \theta+ i\sin\theta$. It is named after the 18th-century mathematician Leonhard Euler.

Background

Euler's formula is a fundamental tool used when solving problems involving complex numbers and/or trigonometry. Euler's formula replaces "cis", and is a superior notation, as it encapsulates several nice properties:

De Moivre's Theorem

De Moivre's Theorem states that for any real number $\theta$ and integer $n$, $(\cos(\theta) + i\sin(\theta))^n = (e^{i\theta})^n = e^{in\theta} = \cos(n\theta) + i\sin(n\theta)$.

Sine/Cosine Angle Addition Formulas

Start with $e^{i(\alpha + \beta)} = (e^{i\alpha})(e^{i\beta})$, and apply Euler's forumla both sides:

$\cos(\alpha + \beta) + i \sin(\alpha + \beta) = (\cos\alpha + i\sin\alpha)(\cos\beta + i\sin\beta).$

Expanding the right side gives

$(\cos\alpha\cos\beta - \sin\alpha\sin\beta) + i(\cos\alpha\sin\beta + \sin\alpha\cos\beta).$

Comparing the real and imaginary terms of these expressions gives the sine and cosine angle-addition formulas:

$\cos(\alpha+\beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta$

$\sin(\alpha+\beta) = \cos\alpha\sin\beta + \sin\alpha\cos\beta$

Geometry on the complex plane

Other nice properties

A special, and quite fascinating, consequence of Euler's formula is the identity $e^{i\pi}+1=0$, which relates five of the most fundamental numbers in all of mathematics: e, i, pi, 0, and 1.

Proof 1

The proof of Euler's formula can be shown using the technique from calculus known as Taylor series.

We have the following Taylor series:

$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots=\sum_{k=0}^{\infty}\frac{x^k}{k!}$

$\sin(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots=\sum_{k=0}^{\infty}(-1)^{k}\frac{x^{2k+1}}{(2k+1)!}$

$\cos(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots=\sum_{k=0}^{\infty}(-1)^{k}\frac{x^{2k}}{(2k)!}$

The key step now is to let $x=i\theta$ and plug it into the series for $e^x$. The result is Euler's formula above.

Proof 2

Define $z=\cos{\theta}+i\sin{\theta}$. Then $\frac{dz}{d\theta}=-\sin{\theta}+i\cos{\theta}=iz$, $\implies \frac{dz}{z}=id\theta$

$\int \frac{dz}{z}=\int id\theta$

$\ln{|z|}=i\theta+c$

$z=e^{i\theta+c}$; we know $z(0)=1$, so we get $c=0$, therefore $z=e^{i\theta}=\cos{\theta}+i\sin{\theta}$.

See Also

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