Difference between revisions of "Euler's identity"
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− | Euler's | + | '''Euler's Formula''' is <math>e^{i\theta}=\cos \theta+ i\sin\theta</math>. 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: | |
− | <math>\sin( | + | ===De Moivre's Theorem=== |
+ | [[De Moivre's Theorem]] states that for any [[real number]] <math>\theta</math> and integer <math>n</math>, | ||
+ | <math>(\cos(\theta) + i\sin(\theta))^n = (e^{i\theta})^n = e^{in\theta} = \cos(n\theta) + i\sin(n\theta)</math>. | ||
− | + | ===Sine/Cosine Angle Addition Formulas=== | |
− | ( | + | Start with <math>e^{i(\alpha + \beta)} = (e^{i\alpha})(e^{i\beta})</math>, and apply Euler's forumla both sides: |
− | + | <math> | |
+ | \cos(\alpha + \beta) + i \sin(\alpha + \beta) = (\cos\alpha + i\sin\alpha)(\cos\beta + i\sin\beta).</math> | ||
− | 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 | + | Expanding the right side gives |
+ | |||
+ | <math> | ||
+ | (\cos\alpha\cos\beta - \sin\alpha\sin\beta) + i(\cos\alpha\sin\beta + \sin\alpha\cos\beta). | ||
+ | </math> | ||
+ | |||
+ | Comparing the real and imaginary terms of these expressions gives the sine and cosine angle-addition formulas: | ||
+ | |||
+ | <math> | ||
+ | \cos(\alpha+\beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta | ||
+ | </math> | ||
+ | |||
+ | <math> | ||
+ | \sin(\alpha+\beta) = \cos\alpha\sin\beta + \sin\alpha\cos\beta | ||
+ | </math> | ||
+ | |||
+ | ===Geometry on the complex plane=== | ||
+ | |||
+ | ===Other nice properties=== | ||
+ | |||
+ | 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. | ||
+ | |||
+ | ==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: | ||
+ | |||
+ | <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. | ||
+ | |||
+ | ==Proof 2== | ||
+ | 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> | ||
+ | |||
+ | <math>\int \frac{dz}{z}=\int id\theta</math> | ||
+ | |||
+ | <math>\ln{|z|}=i\theta+c</math> | ||
+ | |||
+ | <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 == | ||
+ | *[[Complex numbers]] | ||
+ | *[[Trigonometry]] | ||
*[[Power series]] | *[[Power series]] | ||
*[[Convergence]] | *[[Convergence]] | ||
+ | |||
+ | [[Category:Complex numbers]] |
Latest revision as of 22:17, 4 January 2021
Euler's Formula is . It is named after the 18th-century mathematician Leonhard Euler.
Contents
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 and integer , .
Sine/Cosine Angle Addition Formulas
Start with , and apply Euler's forumla both sides:
Expanding the right side gives
Comparing the real and imaginary terms of these expressions gives the sine and cosine angle-addition formulas:
Geometry on the complex plane
Other nice properties
A special, and quite fascinating, consequence of Euler's formula is the identity , 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:
The key step now is to let and plug it into the series for . The result is Euler's formula above.
Proof 2
Define . Then ,
; we know , so we get , therefore .