# Complex-Order Models: A System Identification Point of View
## 2 Operators of Complex Order  
### Differintegral of Complex Order  
The integral operator of complex order \(\alpha+j\beta\) (for some positive real \(\alpha\) and real \(\beta\), positive or negative) of an arbitrary signal \(f:\mathbb{R}^{+}\to\mathbb{C}\) is defined as:  
\[{}_{0}\mathcal{I}_{t}^{\alpha+j\beta}f\coloneqq\frac{1}{\Gamma(\alpha+j\beta )}\int_{0}^{t}f(\tau)(t-\tau)^{\alpha-1+j\beta}d\tau. \tag{1}\]  
As usual, the corresponding complex-order derivative of order \(\alpha+j\beta\) (with \(\alpha>0\) and \(\beta\in\mathbb{R}\)) is defined as:  
\[{}_{0}\mathcal{D}_{t}^{\alpha+j\beta}f\coloneqq\left(\frac{d}{dt}\right)^{n}{} _{0}\mathcal{I}_{t}^{n-\alpha-j\beta}f\, \tag{2}\]  
where \(n\) is the smallest integer greater than \(\alpha\). Similar to the real-order differintegrals, by neglecting the initial conditions, both operators can be uniformly represented in the Laplace domain by complex powers of the Laplace variable \(s\): \(\mathcal{L}\{_{0}\mathcal{I}_{t}^{\alpha+j\beta}f\}(s)=\frac{1}{s^{\alpha+j \beta}}\,F(s)\), and \(\mathcal{L}\{_{0}\mathcal{D}_{t}^{\alpha+j\beta}f\}(s)=s^{\alpha+j\beta}\,F(s)\), where \(F(s)=\mathcal{L}\{f\}(s)\).  
Properties in the Frequency Domain  
The complex-order differintegral naturally extends and generalizes the frequency-domain properties of the conventional differintegral operator of real order. In particular:  
\[(j\omega)^{\alpha+j\beta}=e^{\alpha\ln\omega-\beta\frac{\pi}{2}}e^{j(\beta\ln \omega+\alpha\frac{\pi}{2})}\, \tag{3}\]  
implying that \(|(j\omega)^{\alpha+j\beta}|_{dB}=20\alpha\log\omega-20\beta\frac{\pi}{2}\log e\), and \(\arg(j\omega)^{\alpha+j\beta}=\alpha\frac{\pi}{2}+\beta\ln\omega\).  
The appeal of complex-order differintegrals is clearly visible from these expressions (as illustrated also by Figure 1): similarly to the real-order differintegral, the complex-order differintegral has the amplitude characteristic of constant slope, which is completely determined by the real part of the differentiation order. Contrary to the real-order case, there is an additional offset, which is solely influenced by the imaginary part of the differentiation order. Again, similarly to the real-order case, the phase characteristic has a constant offset equal to \(\alpha\frac{\pi}{2}\) rad; yet, in addition, it has a constant slope equal to \(\beta\) rad/decade (in the logarithmic scale).  
Figure 1: Amplitude (**left**) and phase (**right**) characteristics of the complex-order differintegral \(s^{\alpha+j\beta}\).  
Properties in the Time Domain  
Let us introduce \(A(\alpha,\beta)\) and \(B(\alpha,\beta)\) as the real and imaginary parts, respectively, of the gamma function of the complex argument \(\frac{1}{\Gamma(\alpha+j\beta)}=A(\alpha,\beta)+jB(\alpha,\beta)\). The kernel of the complex-order integral operator can be represented as:  
\[k_{\alpha,\beta}(\tau) =\frac{1}{\Gamma(\alpha+j\beta)}\,\tau^{\alpha-1+j\beta}\] \[=(A(\alpha,\beta)+jB(\alpha,\beta))\tau^{\alpha-1}(\cos(\beta \ln\tau)+j\sin(\beta\ln\tau))\]  
with the real and imaginary parts (also depicted in Figure 2):  
\[\Re\{k_{\alpha,\beta}(\tau)\} =\tau^{\alpha-1}(A(\alpha,\beta)\cos(\beta\ln\tau)-B(\alpha,\beta )\sin(\beta\ln\tau))\, \tag{4}\] \[\Im\{k_{\alpha,\beta}(\tau)\} =\tau^{\alpha-1}(A(\alpha,\beta)\sin(\beta\ln\tau)+B(\alpha,\beta )\cos(\beta\ln\tau)). \tag{5}\]