Fractional calculus

Remarks

For a positive integer $\alpha = n$, $I_a^n f$ is the $n$-fold iterated integral of $f$ from $a$, and $D_a^n f$ is the ordinary higher derivative $f^{(n)}$. Example: with $a = 0$ and $f(t) = 1$, $I_0^{1/2} f(x) = \frac{1}{\Gamma(1/2)} \int_0^x (x - t)^{-1/2} \, dt = \frac{2\sqrt{x}}{\sqrt{\pi}}$. Non-example: for $\alpha = 0$ the integral formula is not defined, since $\Gamma(0)$ is not defined; the operator $I_a^0$ is set to be the identity by convention. The fractional integrals satisfy $I_a^\alpha I_a^\beta = I_a^{\alpha + \beta}$ for $\alpha, \beta > 0$. Other fractional derivatives, such as the Caputo derivative and the Grünwald–Letnikov derivative, agree with $D_a^\alpha$ on suitable classes of functions.