Fractional derivative ($D_{0+}^{\alpha }$)

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 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.