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University Math Lab: Systems & Structures FREE
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Euler's Formula & Complex Exponentials
Learning Objectives
- Apply Euler's formula
- Express complex numbers in exponential form
- Derive trig identities
Key Concepts
Euler's Formula
e^(iθ) = cos(θ) + i·sin(θ)
Euler's Identity
e^(iπ) + 1 = 0
Theory
**Euler's Formula:** e^(iθ) = cos(θ) + i·sin(θ)
**Consequences:**
- cos(θ) = (e^(iθ) + e^(-iθ))/2
- sin(θ) = (e^(iθ) - e^(-iθ))/(2i)
- |e^(iθ)| = 1 for all θ
**Euler's Identity:** e^(iπ) + 1 = 0 (connects e, i, π, 1, 0)
**Complex exponential:** z = re^(iθ) is the polar form of z.
Examples
Express 1+i in exponential form
Solution: |z| = √2, θ = π/4. So z = √2·e^(iπ/4)