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A fractional derivative extends differentiation to noninteger orders, but a choice of definition and endpoint data is part of the operator. For $0<\alpha <1$, the left Riemann-Liouville version iswhen the integral and derivative exist. Here $\Gamma$ is the gamma function. The value depends on a whole interval of earlier values rather than only on values arbitrarily near $t$, making such operators useful in models with memory.^[ref]

For $\beta >-1$ and $t>0$, direct use of the beta function gives $D_{0+}^{\alpha }{t}^{\beta }=\Gamma (\beta +1)t^{\beta -\alpha }/\Gamma (\beta +1-\alpha )$, with the usual reciprocal-gamma interpretation at poles. In particular, $D_{0+}^{1/2}1=1/\sqrt{\pi t}$. The alternative Caputo definition integrates $f^{\prime }$ instead and sends a constant to zero. These are different operators, even though both recover familiar derivatives in appropriate integer-order limits.

Takeshi Matsumoto studied the multifractal measure of the Feigenbaum attractor with Uriel Frisch and Konstantin Khanin. Their analysis related power-law tails of fractional derivatives of an integrated invariant measure to its spectrum of local singularities.^[ref]

This supplies a way to detect nonuniform scaling from a transformed observable, rather than assigning the entire attractor one local exponent. The paper distinguishes its heuristic theoretical steps from high-accuracy numerical checks.

Uriel Frisch studied nonuniform scaling of invariant measures with Konstantin Khanin and Takeshi Matsumoto. Their work on the Feigenbaum attractor related the statistics of fractional derivatives to the local singularity structure of a multifractal measure.^[ref]

A single fractal dimension need not describe how probability concentrates differently from point to point. The derivative-based test offers another way to probe that spatially varying scaling, with the paper explicitly comparing heuristic arguments and precise numerical calculations.