AcademyComplex Form
Academy
Euler Formula
Level 1 - Math I (Physics) topic page in Complex Form.
Euler's Formula
Euler's formula is one of the most beautiful results in mathematics, connecting exponential and trigonometric functions.
The Formula
Euler's Formula
\[e^{i\theta} = \cos\theta + i\sin\theta\]
This remarkable equation shows that complex exponentials are intimately related to circular motion.
Derivation
Using the Taylor series expansions:
Exponential Series
\[e^{i\theta} = 1 + i\theta + \frac{(i\theta)^2}{2!} + \frac{(i\theta)^3}{3!} + \cdots\]
Cosine Series
\[\cos\theta = 1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \cdots\]
Sine Series
\[\sin\theta = \theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \cdots\]
Substituting and grouping real and imaginary terms yields \(e^{i\theta} = \cos\theta + i\sin\theta\).
Special Cases
Setting \(\theta = \pi\) gives the famous Euler's identity:
Euler's Identity
\[e^{i\pi} + 1 = 0\]
This connects five fundamental constants: \(e\), \(i\), \(\pi\), \(1\), and \(0\).
Other useful forms:
- \(e^{-i\theta} = \cos\theta - i\sin\theta\)
- \(\cos\theta = \frac{e^{i\theta} + e^{-i\theta}}{2}\)
- \(\sin\theta = \frac{e^{i\theta} - e^{-i\theta}}{2i}\)