Difference between revisions of "Euler's identity"

Latest revision as of 11:19, 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 $\frac{dz}{d\theta}=-\sin{\theta}+i\cos{\theta}=iz$ , $\implies \frac{dz}{z}=id\theta$ $\int \frac{dz}{z}=\int id\theta$ $\ln{|z|}=i\theta+c$ $z=e^{i\theta+c}<math>; we know </math>z(0)=1<math>, so we get </math>c=0<math>, therefore </math>z=e^{i\theta}=\cos{\theta}+i\sin{\theta}$ .

@@ Line 1: / Line 1: @@
-Euler's formula is <math>\displaystyle e^{i\theta}=\cos(\theta)+i\sin(\theta)</math>.  This can be shown using [[Taylor series]] for <math>e^x, \sin(x)</math>, and <math>\cos(x)</math>.
+'''Euler's Formula''' is <math>e^{i\theta}=\cos \theta+ i\sin\theta</math>.   It is named after the 18th-century mathematician [[Leonhard Euler]].
-== Proof ==
+==Background==
-Note that
-<math>e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...=\sum_{k=0}^{\infty}\frac{x^n}{n!}</math>
+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(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-...=\sum_{j=0}^{\infty}(-1)^{j}\frac{x^{2j+1}}{(2j+1)!}</math>
+===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>.
-<math>\cos(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-...=\sum_{i=0}^{\infty}(-1)^{i}\frac{x^{2n}}{(2n)!}</math>
+=== Sine/Cosine Angle Addition Formulas ===
-(where i, j, k are just [[dummy variables]]).
+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>
+Expanding the right side gives
+<cmath>(\cos\alpha\cos\beta - \sin\alpha\sin\beta) + i(\cos\alpha\sin\beta + \sin\alpha\cos\beta).</cmath>
+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>
+<cmath>\sin(\alpha+\beta) = \cos\alpha\sin\beta + \sin\alpha\cos\beta</cmath>
-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).
+===Geometry on the complex plane===
-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.
+===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]], [[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:
+<cmath>e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots=\sum_{k=0}^{\infty}\frac{x^k}{k!}</cmath>
+<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>
+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>
+<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>.
 == See Also ==
+*[[Complex numbers]]
+*[[Trigonometry]]
 *[[Power series]]
 *[[Convergence]]
+[[Category:Complex numbers]]
+{{stub}}

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