Difference between revisions of "Euler's identity"
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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: | 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 | + | ===[[De Moivre's Theorem]]=== |
<math>\displaystyle(\cos(\theta) + i\sin(\theta))^n = (e^{i\theta})^n = e^{in\theta} = \cos(n\theta) + i\sin(n\theta)</math> | <math>\displaystyle(\cos(\theta) + i\sin(\theta))^n = (e^{i\theta})^n = e^{in\theta} = \cos(n\theta) + i\sin(n\theta)</math> |
Revision as of 12:25, 20 July 2006
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
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 of the formula
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.