*3.1. Riemann–Liouville Integral*

The fractional integral of a function *v*(*t*), also called the Riemann–Liouville integral is defined by

$$\mathbf{x}(t) = {}\_{0}l\_{t}^{n}(f(t)) = \int\_{0}^{t} \frac{(t-\tau)^{n-1}}{\Gamma(n)} v(\tau)d\tau \quad 0 < n < 1,\tag{9}$$

where (*n*) is the gamma function.

The fractional integral is in fact a convolution integral, characterized by the impulse response or Kernel, *hn*(*t*), such that:

$$h\_n(t) = \frac{t^{n-1}}{\Gamma(n)}\quad \text{and}\quad \mathbf{x}(t) = h\_n(t) \* \mathbf{v}(t). \tag{10}$$

Using the Laplace transform, we obtain

$$L\{h\_n(t)\} = \frac{1}{\mathbf{s}^{\mathbf{n}'}} \tag{11}$$

where <sup>1</sup> *<sup>s</sup><sup>n</sup>* corresponds to the fractional order integration operator.
