# Basis Functions for a Transient Analysis of Linear Commensurate Fractional-Order Systems
## 5 Time-Domain Response via Basis Functions: Complex \(s^{1}\)-Domain Poles
### Simple Complex Poles  
For the case of simple poles (\(k=1\)), (46) can be converted to the time domain using the following correspondence, published in [25]:  
\[\frac{s^{\delta}}{\left(s^{\alpha}-a\right)^{2}+b^{2}}\triangleq\frac{t^{ \alpha-\delta-1}}{b}\mathrm{Im}(E_{\alpha,\alpha-\delta}(ct^{\alpha})),c=a+ \mathrm{i}b,b\
eq 0, \tag{47}\]  
where \(\mathrm{Im}\) denotes the imaginary part of the complex MLF.  
By successively substituting \(\sigma=0\), \(\sigma=\alpha\) into (47), Formula (46) can be converted to the time domain for \(k=1\) as follows:  
\[\frac{P_{1}+s^{\alpha}Q_{1}}{\left(s^{\alpha}-a\right)^{2}+b^{2}}\triangleq \frac{t^{\alpha-1}}{b}P_{1}\mathrm{Im}(E_{\alpha,\alpha}(ct^{\alpha}))+\frac {t^{-1}}{b}Q_{1}\mathrm{Im}(E_{\alpha,0}(ct^{\alpha})). \tag{48}\]  
Using (6) and (13)-(16), the right-hand side of (48) can be expressed using the functions \(E_{\alpha,\alpha}\):  
\[\frac{P_{1}+s^{\alpha}Q_{1}}{\left(s^{\alpha}-a\right)^{2}+b^{2}}\triangleq t ^{\alpha-1}\Big{[}\Big{(}P_{1}+Q_{1}\frac{a}{b}\Big{)}\mathrm{Im}\{E_{\alpha,\alpha}(ct^{\alpha})\}+Q_{1}\mathrm{Re}\{E_{\alpha,\alpha}(ct^{\alpha})\} \Big{]}. \tag{49}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{\(\frac{s^{\delta}}{\left(s^{\alpha}-a\right)^{2}}\triangleq\frac{t^{ \alpha-\delta-1}}{\left(k-1\right)!}E_{\alpha,\alpha-\delta}^{\left(k-1\right) }(at^{\alpha})\)} \\ \cline{2-5}  & integer order (\(\alpha=1\)) & fractional order \\ \cline{2-5}  & Laplace domain & time domain & Laplace domain & time domain \\ \hline \multirow{2}{*}{\(\delta=0\)} & \(\frac{1}{s-a}\) & \(e^{tt}\) & \(\frac{1}{s^{\alpha}-a}\) & \(t^{\alpha-1}E_{\alpha,\alpha}(at^{\alpha})\) \\ \cline{2-5}  & \(\frac{1}{\left(s-a\right)^{2}}\) & \(\frac{t^{\alpha-1}}{\left(k-1\right)!}e^{tt}\) & \(\frac{1}{\left(s^{\alpha}-a\right)^{4}}\) & \(\frac{t^{\alpha(k-1)}}{\left(k-1\right)!}e^{t-1}E_{\alpha,\alpha}^{\left(k-1 \right)}(at^{\alpha})\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Basic relations between the time and Laplace domains for single and multiple real pole \(a\).  
Equation (49) is the basic formula from which the various correspondences between the Laplace and time domains are derived in Table 3 for simple complex poles. The table shows that the classical waveforms of type \(\exp(at)\sin(bt)\) and \(\exp(at)\cos(bt)\) for IOS correspond to the combinations of basis functions of type \(t^{\alpha-1}\) Im\([E_{\alpha,\alpha}\) (\(ct^{\alpha}\))]\) and \(t^{\alpha-1}\) Re\([E_{\alpha,\alpha}\) (\(ct^{\alpha}\))] for FOS. In fact, these are the imaginary and real parts of fractional exponential function (26) with a complex argument.  
Interesting results can be obtained for the Laplace transforms without a damping factor, i.e., with the parameter \(a=0\), which lead to undamped oscillations of type \(\sin(bt)\) and \(\cos(bt)\) for IOS. The corresponding fractional goniometric functions are \(\sin_{\alpha,\alpha}(bt^{\alpha})\) and \(\cos_{\alpha,\alpha}(bt^{\alpha})\) multiplied by the function \(t^{\alpha-1}\). However, these are modified fractional goniometric functions (36) and (37).