Hyperbolic Tangent

📐 Definition

For $x \in \mathbb{R}$ ,

\tanh x = \frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}

Domain and Codomain

$\tanh$ is analytic on $\mathbb{C}$ except at the poles of $\cosh$ , i.e. $z = \left(k+\tfrac12\right)\pi i$ for $k\in\mathbb{Z}$ . For real inputs, $\tanh:\mathbb{R}\to(-1,1)$ .

⚙️ Key Properties

Parity and Limits

\tanh(-x) = -\tanh x, \qquad \lim_{x\to\infty}\tanh x = 1, \qquad \lim_{x\to-\infty}\tanh x = -1

Derivative

\frac{d}{dx}\tanh x = \operatorname{sech}^2 x = 1 - \tanh^2 x

Exponential Approach to the Plateaus

As $x\to\infty$ , $1-\tanh x \sim 2e^{-2x}$ , and as $x\to-\infty$ , $\tanh x + 1 \sim 2e^{2x}$ .

🎯 Special Cases and Limits

Near the origin, $\tanh x \sim x$ . The principal singularities in the complex plane occur at $z = \left(k+\tfrac12\right)\pi i$ .

Logistic sigmoid $\sigma(x) = 1/(1+e^{-x})$ is an affine rescaling; smooth saturation functions generalize the bounded response.

Usage in Oakfield

Oakfield uses tanh as a bounded, smooth nonlinearity in both operator kernels and visualization:

Analytic warp: the analytic_warp operator has a TANH profile; its gradient uses $1-\tanh^2(x)$ , making it a safe “LEVEL0” warp for smooth bounded deformation.
Remainder: the remainder operator supports a tanh nonlinearity option before differencing, useful for saturating large amplitudes while keeping the response smooth.
Visualization: the GPU phase view uses tanh to compress magnitude into a bounded value channel for display (helps handle large dynamic ranges).

Historical Foundations

📜 Hyperbolic Functions

Hyperbolic functions arose as analogues of trigonometric functions for the geometry of hyperbolas and became standard tools in analysis and differential equations through 18th–19th century development.

🌍 Modern Perspective

$\tanh$ is a canonical smooth saturating nonlinearity and a standard building block in numerical modeling, optimization, and signal processing.

📚 References

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