Exponential Function

📐 Definition

For $x \in \mathbb{R}$ (and more generally $x \in \mathbb{C}$ ), the exponential function is defined by its power series

\exp(x) = \sum_{n=0}^{\infty} \frac{x^n}{n!}

This definition also characterizes $\exp$ as the unique solution of $y' = y$ with $y(0)=1$ .

Domain and Codomain

$\exp$ is an entire function on $\mathbb{C}$ . For real inputs, $\exp(x) \in (0,\infty)$ .

⚙️ Key Properties

Functional Equation

\exp(a+b) = \exp(a)\exp(b)

Derivative and Integral

\frac{d}{dx}\exp(x) = \exp(x), \qquad \int \exp(x)\,dx = \exp(x) + C

Inverse Relationship with the Logarithm

\log(\exp(x)) = x, \qquad x\in\mathbb{R}

On positive reals, $\exp(\log x)=x$ for $x>0$ . In the complex plane, $\exp(\operatorname{Log}(z))=z$ holds on $\mathbb{C}\setminus(-\infty,0]$ , while $\operatorname{Log}(\exp(z))=z$ only within the principal strip $\operatorname{Im}(z)\in(-\pi,\pi]$ .

Complex Exponential (Euler’s Formula)

\exp(ix) = \cos x + i\sin x

🎯 Special Values and Limits

$x$	$\exp(x)$
$0$	$1$
$1$	$e$
$-1$	$e^{-1}$

As $x \to -\infty$ , $\exp(x)\to 0$ , and as $x \to +\infty$ , $\exp(x)\to +\infty$ .

The logarithm is the inverse of $\exp$ on $(0,\infty)$ . The complex exponential generates trigonometric functions via Euler’s formula, and stretched exponentials generalize decay profiles.

Usage in Oakfield

Oakfield uses exp (and its complex variant) in several core runtime paths:

Spectral dissipation: the linear_dissipative operator implements fractional-Laplacian damping by multiplying each Fourier bin by a factor like $\exp(-\nu |k|^\alpha\,dt)$ .
Spectral phase evolution: the dispersion operator applies per-bin phase factors of the form $e^{i\phi(k)}$ via cexp.
Envelopes and kernels: Gaussian/Gabor stimuli and Gaussian-shaped filters use exponential envelopes (e.g. stimulus_gaussian, stimulus_gabor, and sieve’s Gaussian tap generation).
Stochastic processes: the stochastic_noise operator uses exponential decay terms (including stretched-exponential-style decays) when updating its internal noise state.
Diagnostics smoothing: runtime phase-coherence diagnostics use $\exp(-dt/\tau)$ when computing exponential moving averages (EMA) and lock detection.

Historical Foundations

📜 From Logarithms to $e$

The modern exponential function crystallized through the development of logarithms and continuous compounding, with Euler’s work establishing the base- $e$ exponential as the natural analytic object.

🌍 Modern Perspective

Today, $\exp$ is foundational across analysis and applied modeling, providing the canonical map from additive to multiplicative structure.

📚 References

NIST Digital Library of Mathematical Functions (DLMF), §4
Abramowitz & Stegun, Handbook of Mathematical Functions