# Basis Functions for a Transient Analysis of Linear Commensurate Fractional-Order Systems
## 5 Time-Domain Response via Basis Functions: Complex \(s^{1}\)-Domain Poles
### Multiple Complex Poles  
To convert partial fraction (46) into the time domain, one can proceed in a similar way as for the simple pole, but this time using the relation we published in [25]:  
\[ \begin{array}{l}\frac{e^{\delta}}{[(s^{\alpha}-a)^{2}+b^{2}]^{\frac{\alpha} {2}}}\\ -\frac{A_{1}}{0!}t^{\alpha-\delta-1}\text{Im}(E_{\alpha,\alpha-\delta}(ct^{ \alpha}))-\frac{A_{2}}{1!}t^{2\alpha-\delta-1}\text{Re}\Big{(}E_{\alpha,\alpha -\delta}^{(1)}(ct^{\alpha})\Big{)}\\ +\frac{A_{2}}{0!}t^{3\alpha-\delta-1}\text{Im}\Big{(}E_{\alpha,\alpha-\delta}^ {(2)}(ct^{\alpha})\Big{)}+\frac{A_{4}}{3!}t^{4\alpha-\delta-1}\text{Re}\Big{(} E_{\alpha,\alpha-\delta}^{(3)}(ct^{\alpha})\Big{)}\\ -\frac{A_{5}}{4!}t^{5\alpha-\delta-1}\text{Im}\Big{(}E_{\alpha,\alpha-\delta}^ {(4)}(ct^{\alpha})\Big{)}-\frac{A_{6}}{5!}t^{6\alpha-\delta-1}\text{Re}\Big{(} E_{\alpha,\alpha-\delta}^{(5)}(ct^{\alpha})\Big{)}\\ =t^{\alpha-1}\sum\limits_{i=0}^{\left\{\frac{k-1}{2}\right\}}(-1)^{i+1}A_{2i+ 1}\frac{t^{2i-\delta}}{\left(2i\right)!}\text{Im}\Big{(}E_{\alpha,\alpha- \delta}^{(2)}(ct^{\alpha})\Big{)}+t^{\alpha-1}\sum\limits_{i=0}^{\left\{\frac{ k-1}{2}\right\}}(-1)^{i+1}A_{2(i+1)}\frac{t^{\left(2i+1\right)\alpha-\delta}}{ \left(2i+1\right)!}\text{Re}\Big{(}E_{\alpha,\alpha-\delta}^{(2i+1)}(ct^{ \alpha})\Big{)}\end{array}  \tag{50}\]  
where \([x]\) denotes the integer part of \(x\), and  
\[A_{i}=2(-1)^{i}\binom{2k-i-1}{k-i}(2b)^{-(2k-i)},i=1,2,\ldots,k. \tag{51}\]  
Table 4 summarizes the Laplace transforms and their time representations (50) for the cases \(\delta=0\), \(\delta=\alpha\), and for the case of zero damping (\(a=0\)). The responses for \(\delta=\alpha\), which are expressed using the derivatives of the \(E_{\alpha,0}\) function according to (50), are adjusted to forms that use the \(E_{\alpha,\alpha}\) function via identity (9). \begin{table}
\begin{tabular}{c c c c c} \hline  & & \(\frac{e^{\delta}}{(s^{\alpha}-a)^{2}+b^{2}}\sum\limits_{i=0}^{\frac{\alpha- \delta-1}{2}}\text{Im}\big{(}E_{\alpha,\alpha-\delta}(ct^{\alpha})\big{)}\) & \\ \cline{2-5}  & \multicolumn{2}{c}{integer order (\(\alpha=1\))} & \multicolumn{2}{c}{fractional order} \\ \cline{2-5}  & Laplace domain & time domain & Laplace domain & time domain \\ \hline \multirow{3}{*}{\(\delta=0\)} & \(\frac{b}{(s-a)^{2}+b^{2}}\) & \(e^{at}\sin(bt)\) & \(\frac{b}{(s^{\alpha}-a)^{2}+b^{2}}\) & \(t^{\alpha-1}\text{Im}(E_{\alpha,\alpha}(ct^{\alpha}))\) \\ \cline{2-5}  & \(\frac{b}{s^{2}+b^{2}}\) & \(\sin(bt)\) & \(\frac{b}{s^{2}+b^{2}}\) & \(t^{\alpha-1}\text{Im}(E_{\alpha,\alpha}(\text{i}bt^{\alpha}))=t^{\alpha-1}\sin _{\alpha,\alpha}(bt^{\alpha})\) \\ \cline{2-5}  & \(\frac{s}{(s-a)^{2}+b^{2}}\) & \(e^{at}[\frac{\delta}{\xi}\sin(bt)+\cos(bt)]\) & \(\frac{s^{\alpha}}{(s^{\alpha}-a)^{2}+b^{2}}\) & \(t^{\alpha-1}\text{Im}\big{(}\frac{\xi}{\xi}E_{\alpha,\alpha}(ct^{\alpha})\big{)} =t^{\alpha-1}\big{[}\frac{\alpha}{\xi}\text{Im}\big{(}E_{\alpha,\alpha}(ct^{ \alpha})\big{)}+\text{Re}(E_{\alpha,\alpha}(ct^{\alpha})\big{)}\big{]}\) \\ \cline{2-5}  & \(\frac{s}{s^{2}+b^{2}}\) & \(\cos(bt)\) & \(\frac{s^{\alpha}}{s^{2}+b^{2}}\) & \(t^{\alpha-1}\text{Re}(E_{\alpha,\alpha}(\text{i}bt^{\alpha}))=t^{\alpha-1} \cos_{\alpha,\alpha}(bt^{\alpha})\) \\ \hline \multirow{3}{*}{\(\delta=\alpha\)} & \(\frac{s-a}{(s-a)^{2}+b^{2}}\) & \(e^{at}\cos(bt)\) & \(\frac{s^{\alpha}-a}{(s^{\alpha}-a)^{2}+b^{2}}\) & \(t^{\alpha-1}\text{Re}(E_{\alpha,\alpha}(ct^{\alpha}))\) \\ \cline{2-5}  & \(\frac{s}{(s-a)^{2}+b^{2}}\) & \(e^{at}\cos(bt)\) & \(\frac{s^{\alpha}-a}{(s^{\alpha}-a)^{2}+b^{2}}\) & \(t^{\alpha-1}\text{Re}(E_{\alpha,\alpha}(ct^{\alpha}))\) \\ \hline \end{tabular}
\end{table} 
Table 3: Basic relations between the time and Laplace domains for a simple pair of complex conjugate poles \(c=a+\text{i}b\), \(c^{\star}=a-\text{i}b\).