Complex derivative

Solution 1:

Let $\alpha$ be a complex number with $\mathrm{Re}(\alpha) \geq 0$. Then the Riemann-Liouville fractional derivative of order $\alpha$ of a function $f(t)$ is given by $$_a D^\alpha_t f(t) = \frac{1}{\Gamma(n - \alpha)} \frac{d^n}{dx^n} \int_a^x \frac{f(\tau)}{(t - \tau)^{n-\alpha-1}} ~d\tau,$$ where $n = [\mathrm{Re}(\alpha)] + 1$ and $[\mathrm{Re}(\alpha)]$ is the integer part of $\mathrm{Re}(\alpha)$. Note that this definition indeed makes sense for complex $\alpha$ with positive real part, and in particular, $$_a D^i_t f(t) = \frac{1}{\Gamma(1 - i)} \frac{d}{dx} \int_a^x \frac{f(\tau)}{(t - \tau)^{-i}} ~d\tau.$$