Euler’s Formula

Euler’s formula is a fundamental equation connecting the exponential function to trigonometric functions through complex numbers.

Statement

$$ e^{i\theta} = \cos\theta + i\sin\theta $$

where:

• $e$ is Euler’s number (≈ 2.71828)
• $i = \sqrt{-1}$ is the imaginary unit
• $\theta$ is a real number (angle in radians)

Euler’s Identity

Setting $\theta = \pi$ yields Euler’s identity, often called the most beautiful equation in mathematics:

$$ e^{i\pi} + 1 = 0 $$

This single equation relates five fundamental constants: $e$, $i$, $\pi$, 1, and 0.

Derivation

Euler’s formula can be derived by comparing Taylor series:

$$ e^{i\theta} = 1 + i\theta + \frac{(i\theta)^2}{2!} + \frac{(i\theta)^3}{3!} + \cdots $$

Separating real and imaginary parts yields the Taylor series for cosine and sine.

Consequences

Complex Exponential Representation

Any complex number can be written in polar form:

$$ z = r e^{i\theta} = r(\cos\theta + i\sin\theta) $$

where $r = |z|$ and $\theta = \arg(z)$.

Trigonometric Identities

$$ \cos\theta = \frac{e^{i\theta} + e^{-i\theta}}{2} $$$$ \sin\theta = \frac{e^{i\theta} - e^{-i\theta}}{2i} $$

De Moivre’s Theorem

$$ (\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta) $$

Applications

• Fourier analysis: Complex exponentials as basis functions
• Signal processing: Phasors and frequency analysis
• Quantum mechanics: Wave functions
• Electrical engineering: AC circuit analysis

See Also

• Fourier transform