Euler's Formula

In 1748 Euler wrote eⁱᵖ + 1 = 0 - five fundamental constants in one equation. But this isn't just notation: eⁱˣ = cos x + i·sin x is the mathematical core of the FFT, without which there would be no MP3, JPEG, LTE, or modern signal processing. Every numpy.fft.fft() call on servers around the world uses Euler's formula billions of times per second.

• **FFT and DFT:** eⁱˣ is the mathematical engine of the Fast Fourier Transform; every spectral analysis of audio or images passes through it
• **Phasors in electrical engineering:** voltage V(t) = V₀·cos(ωt+φ) is written as Re(V₀·eⁱᵠ·eⁱᵒ᷊ᵗ); circuit analysis becomes complex number arithmetic
• **Quantum mechanics:** wave functions ψ(x,t) = Ae^(i(kx−ωt)) are built on Euler's formula at the foundation of quantum theory

Предварительные знания

• Hyperbolic Functions

Deriving Euler's Formula via Taylor Series

The Taylor expansion of eˣ at x = 0 is eˣ = 1 + x + x²/2! + x³/3! + ⋯. Substituting ix for x and using i² = −1, i³ = −i, i⁴ = 1, the even terms become the cosine series and the odd terms become the sine series.

**Euler's formula:** eⁱˣ = cos x + i·sin x Inverse: cos x = Re(eⁱˣ) = (eⁱˣ + e⁻ⁱˣ)/2 sin x = Im(eⁱˣ) = (eⁱˣ − e⁻ⁱˣ)/(2i)

Geometrically, eⁱˣ is a point on the unit circle at angle x from the real axis. Multiplying a complex number z by eⁱˣ rotates z by angle x. This makes complex numbers and Euler's formula the natural language for describing rotations.

Euler's Identity and Complex Form of Trig Functions

Setting x = π in Euler's formula gives eⁱᵖ = cos π + i·sin π = −1 + 0·i = −1. Adding 1 to both sides: eⁱᵖ + 1 = 0. This is Euler's identity - one of the most widely known equations in mathematics: five fundamental constants (e, i, π, 1, 0) in a single formula.

**Euler's identity:** eⁱᵖ + 1 = 0 **Complex form of trig functions:** cos x = (eⁱˣ + e⁻ⁱˣ) / 2 sin x = (eⁱˣ − e⁻ⁱˣ) / (2i)

De Moivre's Theorem and DFT Applications

De Moivre's theorem follows directly from Euler's formula: (eⁱˣ)ⁿ = eⁱⁿˣ, so (cos x + i·sin x)ⁿ = cos(nx) + i·sin(nx). It's a powerful tool for deriving multiple-angle identities without laborious trigonometric manipulation.

**De Moivre's theorem:** (cos x + i·sin x)ⁿ = cos(nx) + i·sin(nx) Application: expand (cos x + i·sin x)² = cos²x − sin²x + 2i·sin x·cos x Read off double-angle formulas: Re: cos 2x = cos²x − sin²x Im: sin 2x = 2·sin x·cos x

FFT (Fast Fourier Transform) computes the DFT in O(N log N) time by exploiting the symmetry of Euler's formula at evenly spaced angles. It underlies audio processing, image compression (JPEG), wireless communication (LTE/5G), and countless ML preprocessing pipelines.

Key Ideas

• **eⁱˣ = cos x + i·sin x**: derived from Taylor series; |eⁱˣ| = 1, multiplying by eⁱˣ is a rotation
• **Euler's identity:** eⁱᵖ + 1 = 0; **inverse formulas:** cos x = (eⁱˣ+e⁻ⁱˣ)/2, sin x = (eⁱˣ−e⁻ⁱˣ)/(2i)
• **De Moivre's theorem:** (cos x + i·sin x)ⁿ = cos(nx) + i·sin(nx) - derives multiple-angle formulas instantly
• **DFT/FFT:** signal = sum of eⁱᵒ᷊ₙᵗ components; |X[k]| is amplitude, arg(X[k]) is phase of each frequency

Related Topics

Euler's formula unifies trigonometry, analysis, and applications:

• Hyperbolic Functions — cosh x = cos(ix), sinh x = −i·sin(ix) - hyperbolic functions are trig functions with an imaginary argument
• Trigonometry in Graphics and Signals — FFT is a direct application of Euler's formula to spectral analysis; RoPE uses eⁱˣ for positional encoding in transformers
• Trigonometry in Interviews — Rotating a vector by angle θ in the complex plane is multiplication by eⁱᶿ - a geometric shortcut for interview problems

Вопросы для размышления

• Why is the identity eⁱᵖ + 1 = 0 considered significant? What is mathematically special about five fundamental constants being connected this way?
• Euler's formula turns trig identities into algebra of exponents: sin²x + cos²x = 1 becomes eⁱˣ·e⁻ⁱˣ = 1. Derive the angle addition formula cos(a+b) from this.
• FFT runs in O(N log N) instead of O(N²). How is this connected to Euler's formula? Hint: think about what symmetry e^(2πi·k·n/N) has when N is even.

Связанные уроки

• ca-01