Euler's Formula

Euler's Formula & Identity

Mathematical Definition

Euler's formula is defined for any real number θ as:

In rectangular form, this means the complex number e^iθ has:

• Real part: Re(e^iθ) = cos(θ)
• Imaginary part: Im(e^iθ) = sin(θ)
• Magnitude: |e^iθ| = √(cos²θ + sin²θ) = 1

Because its magnitude is always 1 regardless of θ, e^iθ always lies on the unit circle in the complex plane. As θ increases from 0 to 2π, the point completes one full counterclockwise revolution.

Euler's Identity

Substituting θ = π into Euler's formula gives:

This single equation unites the five most important constants in mathematics: e (Euler's number, the base of natural logarithms), i (the imaginary unit, √−1), π (the ratio of a circle's circumference to its diameter), 1 (the multiplicative identity), and 0 (the additive identity).

Derivation via Taylor Series

Euler's formula can be derived by substituting iθ into the Taylor series expansion of e^x:

Polar Form of Complex Numbers

Any complex number z = a + bi can be written in polar form using Euler's formula:

where r = |z| = √(a² + b²) is the magnitude and θ = arg(z) = atan2(b, a) is the argument (angle). This polar representation makes multiplication of complex numbers especially elegant: multiplying two complex numbers multiplies their magnitudes and adds their angles.

Key Concepts

• Unit circle: e^iθ always has magnitude 1, so it lies on the unit circle in the complex plane for all real θ.
• Periodicity: e^iθ is periodic with period 2π: e^i(θ+2π) = e^iθ. One full revolution of the unit circle corresponds to an increase of 2π in θ.
• Complex conjugate: e^−iθ = cos(θ) − i·sin(θ) is the reflection of e^iθ across the real axis. Together they give cos(θ) = (e^iθ + e^−iθ)/2 and sin(θ) = (e^iθ − e^−iθ)/(2i).
• De Moivre's theorem: (e^iθ)ⁿ = e^inθ, or equivalently (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ). This is a direct consequence of Euler's formula.

Historical Context

Leonhard Euler (1707–1783), the Swiss mathematician after whom the formula is named, was one of the most prolific mathematicians in history. He introduced the notation e for the base of natural logarithms and i for √−1. Euler published the formula in its modern form in his 1748 work Introductio in analysin infinitorum, though special cases had been noted by Roger Cotes in 1714.

The specific form known as Euler's Identity (e^iπ + 1 = 0) was highlighted as a special case of the formula. In a 1988 poll of readers of The Mathematical Intelligencer, Euler's Identity was voted the "most beautiful theorem in mathematics." Physicist Richard Feynman called it "the most remarkable formula in mathematics."

Real-world Applications

• Signal processing & Fourier analysis: The Discrete Fourier Transform (DFT) and Fast Fourier Transform (FFT) are built entirely on Euler's formula. Every sinusoidal component is represented as a complex exponential e^iωt.
• Electrical engineering: AC circuit analysis uses phasors—complex numbers in polar form—to represent sinusoidal voltages and currents. Impedance calculations become simple multiplications of complex exponentials.
• Quantum mechanics: Wave functions in quantum mechanics are expressed as complex exponentials e^ikx−iωt, and Euler's formula connects the oscillatory behavior of these waves to observable probabilities.
• Control theory: The Laplace and Z-transforms, fundamental tools for analyzing stability and dynamic systems, rely on complex exponentials. The unit circle in the Z-domain corresponds directly to Euler's formula.
• Computer graphics: Rotations in 2D can be represented by multiplying by e^iθ. In 3D, quaternions — a generalization of complex numbers — use the same exponential map to represent smooth rotations without gimbal lock.

Related Concepts

• Taylor Series — the derivation of Euler's formula relies on Taylor expansions of eˣ, sin(x), and cos(x)
• Fourier Transform — decomposes signals into complex exponentials using Euler's formula
• Harmonic Oscillator — solutions to the oscillator ODE are naturally expressed as e^iωt
• Complex Number Visualization — Euler's formula is the bridge between rectangular and polar forms