04 // Interactive Formula Compendium // CALCULUS-ALGEBRAe^(iθ) = cos(θ) + i · sin(θ) ⇒ e^(iπ) + 1 = 0
Euler's Formula & Euler's Identity — Complex Exponential Reference Sheet
Euler's formula states that for any real angle θ in radians, e^(iθ) = cos(θ) + i sin(θ); substituting θ = π rad yields Euler's Identity e^(iπ) + 1 = 0, linking the five fundamental mathematical constants e, i, π, 1, and 0.
Formula Reference
Main Formula
e^(iθ) = cos(θ) + i · sin(θ) ⇒ e^(iπ) + 1 = 0
Rearranged Forms
z = r · e^(iθ) = r cos(θ) + i · r sin(θ) (Polar Complex Form)
cos(θ) = (e^(iθ) + e^(−iθ)) / 2, sin(θ) = (e^(iθ) − e^(−iθ)) / (2i)
(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ) (De Moivre's Theorem)
Variables & Units
| Symbol | Name | Unit |
|---|---|---|
| Re(z) | Real Component r·cos(θ) of r·e^(iθ) | dimensionless |
| r | Complex Modulus / Magnitude (r) | 1 |
| theta | Phase Angle θ (radians) | rad |
Inputs & Parameters
[1]
[rad]
Calculated Result (Real Component r·cos(θ) of r·e^(iθ))
Re(z) =-1dimensionless
Formula Notes & Background
- Write the Maclaurin series for e^(iθ) = Σ (iθ)ᵏ / k! = 1 + iθ − θ²/2! − iθ³/3! + θ⁴/4! + iθ⁵/5! − …
- Group all real even-power terms: (1 − θ²/2! + θ⁴/4! − …) = cos(θ).
- Group all imaginary odd-power terms: i · (θ − θ³/3! + θ⁵/5! − …) = i sin(θ).
Step-by-Step Examples
#1: Evaluate e^(iπ) and verify Euler's Identity.
r = 1θ = π rad
e^(iπ) = cos(π) + i sin(π) = −1 + i(0) = −1
→ e^(iπ) = −1 ⇒ e^(iπ) + 1 = 0
#2: Convert the phasor z = 10 e^(i π/3) into rectangular form a + bi.
r = 10θ = π/3 rad (60°)
z = 10 cos(π/3) + i · 10 sin(π/3) = 10(0.5) + i · 10(√3/2)
→ z = 5 + 8.66025i
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Frequently Asked Questions
Why is Euler's formula indispensable in electrical engineering and signal processing?+
Representing sinusoidal AC signals as phasors V · e^(iωt) turns calculus differential equations (inductors and capacitors) into algebraic multiplication by iω.
What is the magnitude |e^(iθ)| for any real angle θ?+
|e^(iθ)| = √(cos²θ + sin²θ) = 1, meaning e^(iθ) always lies on the unit circle in the complex plane.
How does De Moivre's Theorem follow from Euler's formula?+
By exponent laws, (e^(iθ))ⁿ = e^(inθ), which immediately translates to (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ).