Complex Exponential

📐 Definition

For a real phase variable $\theta$ , the complex exponential is

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

Domain and Codomain

For $\theta \in \mathbb{R}$ , $e^{i\theta}$ lies on the complex unit circle $\{z\in\mathbb{C}:|z|=1\}$ . The map extends to complex arguments $\theta\in\mathbb{C}$ as an entire function.

⚙️ Key Properties

Unit Magnitude and Periodicity (Real Phase)

|e^{i\theta}| = 1, \qquad e^{i(\theta + 2\pi k)} = e^{i\theta}\quad (k\in\mathbb{Z})

Conjugation

\overline{e^{i\theta}} = e^{-i\theta}

Multiplicative Phase Composition

e^{i(\theta_1+\theta_2)} = e^{i\theta_1}e^{i\theta_2}

🎯 Special Values and Limits

$\theta$	$e^{i\theta}$
$0$	$1$
$\pi$	$-1$
$\tfrac{\pi}{2}$	$i$

For small $|\theta|$ , a first-order expansion is $e^{i\theta}\approx 1+i\theta$ .

Real and imaginary parts yield cosine and sine. Fourier series and Fourier transforms use $e^{ikx}$ as basis elements, and standing waves and phase-shifted sinusoids arise from linear combinations of complex exponentials.

Usage in Oakfield

Oakfield uses complex exponentials extensively in spectral and modulation-style code paths:

Spectral phase factors: the dispersion operator multiplies FFT bins by cexp(I * phase) to apply dispersive phase advance.
Mixer modulation modes: FM/PM modes apply $e^{i\theta}$ -style multipliers to complex fields.
Stimulus rotations: several complex-capable stimuli apply an additional unit-magnitude complex rotation when writing outputs.

Historical Foundations

📜 Euler’s Formula

Euler’s identity connecting exponentials and trigonometric functions provides the analytic bridge between additive phase and multiplicative complex rotation.

🌍 Modern Perspective

The complex exponential is the canonical representation of phase and oscillation in spectral methods, wave physics, and signal processing.

📚 References

NIST Digital Library of Mathematical Functions (DLMF), §4
Stein & Shakarchi, Fourier Analysis