= Fractional derivative
{title2=$D_{0+}^{\alpha}$}
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 is
$$
D_{0+}^{\alpha}f(t)=\frac{d}{dt}\left[\frac1{\Gamma(1-\alpha)}\int_0^t(t-s)^{-\alpha}f(s)\,ds\right],
$$
when 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.<NIST Digital Library of Mathematical Functions>[Sections 1.15(vi)–(vii), fractional integrals and derivatives]{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 $f'$ instead and sends a constant to zero. These are different operators, even though both recover familiar derivatives in appropriate integer-order limits.
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