Art of Problem Solving
Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Euler's identity: Difference between revisions

← Older edit
Jam (talk | contribs)
155 edits
mNo edit summary
Charking (talk | contribs)
editor
3,291 edits
m Minor edit
 
(2 intermediate revisions by 2 users not shown)
Line 1: Line 1:
'''Euler's identity''' is <math>e^{i\theta}=\cos \theta+ i\sin\theta</math>.  It is named after the 18th-century mathematician [[Leonhard Euler]].
'''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==
Line 6: Line 6:


===De Moivre's Theorem===
===De Moivre's Theorem===
[[De Moivre's Theorem]] states that for any [[real number]]s <math>\theta</math> and <math>n</math>,  
[[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>.
<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 ===


Start with <math>e^{i(\alpha + \beta)} = (e^{i\alpha})(e^{i\beta})</math>, and apply Euler's forumla both sides:
Start with <math>e^{i(\alpha + \beta)} = (e^{i\alpha})(e^{i\beta})</math>, and apply Euler's forumla both sides:
 
<cmath>\cos(\alpha + \beta) + i \sin(\alpha + \beta) = (\cos\alpha + i\sin\alpha)(\cos\beta + i\sin\beta).</cmath>
<math>
\cos(\alpha + \beta) + i \sin(\alpha + \beta) = (\cos\alpha + i\sin\alpha)(\cos\beta + i\sin\beta).</math>
 
Expanding the right side gives
Expanding the right side gives
 
<cmath>(\cos\alpha\cos\beta - \sin\alpha\sin\beta) + i(\cos\alpha\sin\beta + \sin\alpha\cos\beta).</cmath>
<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:
Comparing the real and imaginary terms of these expressions gives the sine and cosine angle-addition formulas:
 
<cmath>\cos(\alpha+\beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta</cmath>
<math>
<cmath>\sin(\alpha+\beta) = \cos\alpha\sin\beta + \sin\alpha\cos\beta</cmath>
\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===
===Geometry on the complex plane===
Line 36: Line 23:
===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 (constant)| 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]], [[0]], and [[1]].


==Proof 1==
==Proof 1==
Line 43: Line 30:


We have the following Taylor series:
We have the following Taylor series:
 
<cmath>e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots=\sum_{k=0}^{\infty}\frac{x^k}{k!}</cmath>
<math>e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots=\sum_{k=0}^{\infty}\frac{x^k}{k!}</math>
<cmath>\sin(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots=\sum_{k=0}^{\infty}(-1)^{k}\frac{x^{2k+1}}{(2k+1)!}</cmath>
 
<cmath>\cos(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots=\sum_{k=0}^{\infty}(-1)^{k}\frac{x^{2k}}{(2k)!}</cmath>
<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.  
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.  
Line 54: Line 38:
==Proof 2==
==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>
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>
<cmath>\int \frac{dz}{z}=\int id\theta</cmath>
<cmath>\ln{|z|}=i\theta+c</cmath>
<cmath>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}</cmath>.


<math>\int \frac{dz}{z}=\int id\theta</math>
== See Also ==
 
<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 ==
*[[Complex numbers]]
*[[Complex numbers]]
*[[Trigonometry]]
*[[Trigonometry]]
Line 68: Line 50:


[[Category:Complex numbers]]
[[Category:Complex numbers]]
{{stub}}

Latest revision as of 10:19, 15 February 2025

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 $\frac{dz}{d\theta}=-\sin{\theta}+i\cos{\theta}=iz$, $\implies \frac{dz}{z}=id\theta$ \[\int \frac{dz}{z}=\int id\theta\] \[\ln{|z|}=i\theta+c\] \[z=e^{i\theta+c}<math>; we know </math>z(0)=1<math>, so we get </math>c=0<math>, therefore </math>z=e^{i\theta}=\cos{\theta}+i\sin{\theta}\].

See Also

This article is a stub. Help us out by expanding it.

Retrieved from "https://wiki.aopstest.com/wiki/index.php?title=Euler%27s_identity&oldid=242706"