Power Function

📐 Definition

The power function raises an input to a real exponent $p$ . A common real-valued definition uses nonnegative inputs:

f_p(x) = x^p \qquad (x \ge 0)

Domain and Codomain

For real-valued $x^p$ with arbitrary real $p$ , the natural domain is $x\ge 0$ . For integer $p$ , the map extends to all real $x$ .

⚙️ Key Properties

Derivative (Positive Inputs)

\frac{d}{dx}x^p = p\,x^{p-1}, \qquad x>0

Multiplicativity (Positive Inputs)

(ab)^p = a^p b^p, \qquad a>0,\ b>0

Log–Exp Representation

For $x>0$ ,

x^p = \exp(p\log x)

Exponents above one amplify large magnitudes; $0<p<1$ compress them.

🎯 Special Cases and Limits

$p=1$ yields the identity.
$p=2$ gives quadratic growth.
$p=\tfrac12$ is the square root.
Negative exponents give reciprocal powers on $x>0$ : $x^{-p} = 1/x^{p}$ .
For fixed $x>0$ , $\lim_{p\to 0} x^p = 1$ .

Fractional powers are the non-integer cases. Absolute value is commonly combined with powers to handle signed inputs. Logarithm and exponential provide the canonical formulation $x^p=\exp(p\log x)$ .

Usage in Oakfield

Oakfield uses power functions in both pointwise nonlinearities and spectral laws:

Analytic warp (POWER profile) computes power-law gradients/responses for controlled nonlinear deformation.
Remainder (POWER nonlinearity) applies a power transform with an epsilon guard before differencing.
Spectral operators use pow for frequency-dependent laws (e.g. dispersion uses pow(|k-k0|, order), and linear_dissipative uses $|k|^\alpha$ in its damping rate).

Historical Foundations

📜 Exponents and Scaling

Power laws encode multiplicative scaling behavior and have long served as basic models for growth/decay and similarity.

🌍 Modern Perspective

In computation, power functions are used both directly and via log–exp reformulations for numerical stability and generalization.

📚 References

NIST Digital Library of Mathematical Functions (DLMF), §4
Rudin, Real and Complex Analysis