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== | ==Background== | ||
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===De Moivre's Theorem=== | ===De Moivre's Theorem=== | ||
− | + | [[De Moivre's Theorem]] states that for any [[real number]] <math>\theta</math> and integer <math>n</math>, | |
− | <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=== | ===Sine/Cosine Angle Addition Formulas=== | ||
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===Other nice properties=== | ===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 | 0]], and 1. | + | 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 | + | ==Proof 1== |
The proof of Euler's formula can be shown using the technique from [[calculus]] known as [[Taylor series]]. | The proof of Euler's formula can be shown using the technique from [[calculus]] known as [[Taylor series]]. | ||
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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. | 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 == | ||
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*[[Power series]] | *[[Power series]] | ||
*[[Convergence]] | *[[Convergence]] | ||
+ | |||
+ | [[Category:Complex numbers]] |
Latest revision as of 23:17, 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
.