## 2 On the Fractionalisation Problem
Applying the bilateral Laplace transform (BLT),  
\[X(s)=\int\limits_{\mathbf{R}}x(t)e^{-st}\mathrm{d}t, \tag{5}\]  
to (4), we obtain:  
\[\mathcal{L}[x(t)]=s^{\alpha}X(s),\quad Re(s)>0, \tag{6}\]  
generalising a well-known property of the BLT [35]. It is important to remark that (4) and (6) are valid for any positive or negative real order. We will keep the designation \"derivative\" for positive orders. For negative ones, we will use \"anti-derivative\". It must be referred that, contrarily to the primitive, the anti-derivative is simultaneously the left and right inverse of the derivative \(D\ D^{-1}x(t)=D^{-1}\ Dx(t)=x(t)\). Let \(\alpha\) be any real number and \(\varepsilon(t)\) be the Heaviside-unit step function. The causal BLT inverse of \(s^{\alpha}\),  
\[\mathcal{L}^{-1}[s^{\alpha}]=\frac{t^{-\alpha-1}}{\Gamma(-\alpha)}\varepsilon( t),\]  
is the Green function of the derivative operator that, using the convolution property of the BLT, allows us to obtain three integral formulations for the fractional derivatives equivalent to (4). Let \(N\) be the positive integer greater than or equal to \(\alpha\) (\(\alpha\leq N\)). Such derivatives are given by:  
1. (Riemann-)Liouville derivative [30] \[D^{\alpha}x(t)=D^{N}\Bigg{[}\frac{1}{\Gamma(-\alpha+N)}\int\limits_{0}^{\infty }\tau^{N-\alpha-1}x(t-\tau)d\tau\Bigg{]}.\] (7) The usual Riemann-Liouville derivative is a particular case of this one, obtained for functions null outside any interval \([a,b]\subset\mathbf{R}\). 2. Liouville-Caputo derivative [25] \[D^{\alpha}x(t)=\frac{1}{\Gamma(-\alpha+N)}\int\limits_{0}^{\infty}\tau^{N- \alpha-1}x^{(N)}(t-\tau)d\tau.\] (8) The Caputo derivative is a particular case of this one as happens with the RL derivative. 3. Regularized Liouville derivative [40] \[D^{\alpha}x(t)=\frac{1}{\Gamma(-\alpha)}\int\limits_{0}^{\infty}\tau^{-\alpha- 1}\Bigg{[}x(t-\tau)-\sum\limits_{0}^{N-1}\frac{(-)^{m}x^{(m)}(t)}{m!}\tau^{m} \Bigg{]}d\tau,\] (9) where the summation is taken as being null, when \(\alpha<0\). All the derivatives (4), (7) to (9) are equivalent for functions with BLT. As the Fourier transform is obtained from the BLT with a simple variable change \(s=i\omega,\ \omega\in\mathbf{R}\), \(i=\sqrt{-1}\), the above derivatives exist also for functions with Fourier transform [35]. Necessary conditions for the existence of the above derivatives can be given [30,31,41]. It must be highlighted that we must take into account the behaviour of the function \(x(t)\) along the half straight line \(t-nh\), with \(n\in\mathbf{Z}^{+}\). There are several properties exhibited by derivatives (4), (7)-(9) [37]:* Linearity;
* Additivity and Commutativity of the orders;
* Neutral and inverse elements \[D^{\alpha}D^{-\alpha}x(t)=D^{0}x(t)=x(t).\] (10) From (10) we conclude that there is always an inverse element; that is, for every \(\alpha\) there is always the \(-\alpha\) order that we called above the anti-derivative. * Backward compatibility (\(n\in\mathbf{N}\)) If \(\alpha=n\), then: \[D^{\alpha}x(t)+x(t)=\frac{1}{q}x\bigg{(}\frac{t}{q}\bigg{)}\quad t\geq 0,\ q \geq 1,\]0 We obtain this expression repeating the first order derivative. If \(\alpha=-n\), then: \[D^{\alpha}x(t)+x(t)=\frac{1}{q}x\bigg{(}\frac{t}{q}\bigg{)}\quad t\geq 0,\ q \geq 1,\]1 that corresponds to a \(n\)-th repeated summation [41; 42]. For \(n=-1\), we obtain the classic Riemann integral \[D^{\alpha}x(t)+x(t)=\frac{1}{q}x\bigg{(}\frac{t}{q}\bigg{)}\quad t\geq 0,\ q \geq 1,\]2
* The generalized Leibniz rule This rule gives the FD of the product of two functions and assumes the format [30] \[D^{\alpha}x(t)+x(t)=\frac{1}{q}x\bigg{(}\frac{t}{q}\bigg{)}\quad t\geq 0,\ q \geq 1,\]3 (11)  
This approach to fractional derivatives was used to obtain coherent formulations for linear systems, generalising the concept to non-commensurate and tempered ones, in full agreement with classic theory [9]. An interesting application to the fractional logistic equation can be found in [43].