Fractional integrals and fractional derivatives

Fractional integrals are easier to define than fractional derivatives. And for sufficiently smooth functions, you can use the former to define the latter.

The Riemann-Liouville fractional integral starts from the observation that for positive integer n,

$I^n f(x) &\equiv& \int_a^{x} \int_a^{x_1} \cdots \int_a^{x_{n-1}} f(x_n)\,dx_n\, dx_{n-1} \cdots \,dx_1 \\ &=& \frac{1}{(n-1)!} \int_a^x (x-t)^{n-1} f(t)\, dt$

This motivates a definition of fractional integrals

$I^\alpha f(x) = \frac{1}{\Gamma(\alpha)} \int_a^x (x-t)^{\alpha-1} f(t)\, dt$

which is valid for any complex α with positive real part. Derivatives and integrals are inverses for integer degree, and we use this to define fractional derivatives: the derivative of degree n is the integral of degree –n. So if we could define fractional integrals for any degree, we could define a derivative of degree α to be an integral of degree -α.

Unfortunately we can’t do this directly since our definition only converges in the right half-plane. But for (ordinary) differentiable f, we can integrate the Riemann-Liouville definition of fractional integral by parts:

$I^\alpha f(x) = \frac{(x-a)^\alpha}{\Gamma(\alpha+1)} f(a) + I^{\alpha+1} f'(x)$

We can use the right side of this equation to define the left side when the real part of α is bigger than -1. And if f has two ordinary derivatives, we can repeat this process to define fractional integrals for α with real part bigger than -2. We can repeat this process to define the fractional integrals (and hence fractional derivatives) for any degree we like, provided the function has enough ordinary derivatives.

See previous posts for two other ways of defining fractional derivatives, via Fourier transforms and via the binomial theorem.