Difference between revisions of "Euler's identity"
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===De Moivre's Theorem=== | ===De Moivre's Theorem=== | ||
− | [[De Moivre's Theorem]] states that for any [[real number]] | + | [[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>. | ||
Revision as of 11:23, 17 February 2016
Euler's identity 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
.