Fractional Powers

📐 Definition

Fractional powers generalize exponentiation to non-integer exponents, producing intermediate growth or decay behaviors between linear and root or quadratic forms.

For $x>0$, a standard real-valued definition is

$$x^\alpha = \exp(\alpha \log x)$$

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Domain and Codomain

For arbitrary real $\alpha$, the real-valued definition requires $x>0$ (or $x\ge 0$ with conventions at $0$). Extensions to signed values require signed-power conventions or complex outputs.

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⚙️ Key Properties

Derivative (Positive Inputs)

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

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Multiplicativity (Positive Inputs)

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

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Signed-Power Convention (Common)

For signed data, a frequently used real-valued convention is

$$x \mapsto |x|^\alpha\,\operatorname{sgn}(x)$$

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🎯 Special Cases and Limits

• $\alpha=1$ recovers the identity.
• $\alpha=\tfrac12$ is the square root.
• $\alpha=-1$ yields the reciprocal $1/x$ on $x>0$.
• For fixed $x>0$, $\lim_{\alpha\to 0} x^\alpha = 1$.

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🔗 Related Functions

Fractional powers are a subset of the general power function. Logarithm and exponential define $x^\alpha$ via $x^\alpha=\exp(\alpha\log x)$. Absolute value is used to build signed-power conventions that stay real-valued.

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Usage in Oakfield

Oakfield uses fractional powers mainly as parameters in fractional operators and spectral laws:

• Fractional Laplacian damping uses a configurable exponent alpha in linear_dissipative to apply $|k|^\alpha$-style spectral rates.
• Fractional-time kernels use fractional exponents in subordination and colored-noise paths (e.g. stretched-exponential weights and decay exponents).
• Some nonlinearities (e.g. power profiles) can be configured with non-integer exponents where smoothness and near-zero behavior matter.

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Historical Foundations

📜 From Roots to Real Exponents

Fractional exponents generalize roots and powers and become analytically natural through the log–exp identity, which also introduces branch structure in the complex plane.

🌍 Modern Perspective

They are widely used as smooth, tunable shaping functions and as intermediate-scale penalties bridging linear and quadratic behavior.

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📚 References

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