# Basis Functions for a Transient Analysis of Linear Commensurate Fractional-Order Systems
## 6 Basis Functions for Integer- and Fractional-Order Commensurate Systems: One-to-One Correspondence
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Poles** & \multicolumn{2}{c}{**Integer Order (\(\alpha=1\))**} & \multicolumn{2}{c}{**Fractional Order**} \\ \hline  & Basis Functions & Generated from & Basis Functions & Generated from \\  & \(\exp\), \(\sin\), \(\cos\), \(t^{k}\) & \(f(t)=t^{k}e^{at}\) & MLF, \(\sin\), \(\cos\), \(\hat{\e}\), \(t^{ak}\), \(t^{\alpha-1}\) & \(f_{T}(t)=\varepsilon_{k}(t,c;\alpha,\alpha)\) \\ \hline simple real \(a\) & \(e^{at}\) & \(t^{0}e^{at}\) & \(\widetilde{\varepsilon}_{\alpha,\alpha}(a,t)=t^{\alpha-1}E_{\alpha,\alpha}(a ^{\alpha})\) & \(\varepsilon_{0}(t,a;\alpha,\alpha)\) \\ \hline simple complex pair & \(e^{at}\sin(bt)\) & \(\mathrm{Im}(t^{0}e^{ct})\) & \(\mathrm{Im}(\widetilde{e}_{\alpha,\alpha}(c,t))\) & \(\mathrm{Im}(\varepsilon_{0}(t,c;\alpha,\alpha))\) \\ \(c=a+\mathrm{i}b\), \(c^{\alpha}=a-\mathrm{i}b\) & \(e^{at}\cos(bt)\) & \(\mathrm{Re}(t^{0}e^{ct})\) & \(\mathrm{Re}(\widetilde{e}_{\alpha,\alpha}(c,t))\) & \(\mathrm{Re}(\varepsilon_{0}(t,c;\alpha,\alpha))\) \\ \hline simple imaginary pair & \(\sin(bt)\) & \(\mathrm{Im}(\widetilde{t}_{\alpha}b^{bt})\) & \(\mathrm{Im}(\widetilde{e}_{\alpha,\alpha}(\mathrm{i}b,t))=\sin_{\alpha,\alpha} (b,t)\) & \(\mathrm{Im}(\varepsilon_{0}(t,\mathrm{i}b;\alpha,\alpha))\) \\ \(c=+\mathrm{i}b\), \(c^{\alpha}=-\mathrm{i}b\) & \(\cos(bt)\) & \(\mathrm{Re}(\widetilde{t}^{0}e^{bt})\) & \(\mathrm{Re}(\varepsilon_{\alpha,\alpha}(\mathrm{i}b,t))=\cos_{\alpha,\alpha} (b,t)\) & \(\mathrm{Re}(\varepsilon_{0}(t,\mathrm{i}b;\alpha,\alpha))\) \\ \hline multiple real \(a\) & \(t^{k}e^{at}\) & \(t^{k}e^{at}\) & \(t^{ak}t^{ak}t^{ak-1}E_{\alpha,\alpha}^{(k)}(t^{ak})\) & \(\varepsilon_{k}(t,a;\alpha,\alpha)\) \\ \hline multiple complex pair & \(t^{k}e^{at}\sin(bt)\) & \(\mathrm{Im}(t^{k}e^{ct})\) & \(t^{ak}\mathrm{Im}\Big{(}t^{\alpha-1}E_{\alpha,\alpha}^{(k)}(t^{ak})\Big{)}\) & \(\mathrm{Im}(\varepsilon_{k}(t,c;\alpha,\alpha))\) \\ \(c=a+\mathrm{i}b\), \(c^{\alpha}=a-\mathrm{i}b\) & \(t^{k}e^{at}\cos(bt)\) & \(\mathrm{Re}(t^{k}e^{ct})\) & \(t^{ak}\mathrm{Re}\Big{(}t^{ak-1}E_{\alpha,\alpha}^{(k)}(t^{ak})\Big{)}\) & \(\mathrm{Re}(\varepsilon_{k}(t,c;\alpha,\alpha))\) \\ \hline multiple imaginary pair & \(t^{k}\sin(bt)\) & \(\mathrm{Im}(t^{k}e^{bt})\) & \(t^{ak}t^{ak}t^{ak-1}\sin_{\alpha,\alpha}^{(k)}(t^{ak})\) & \(\mathrm{Im}(\varepsilon_{k}(t,\mathrm{i}b;\alpha,\alpha))\) \\ \(c=+\mathrm{i}b\), \(c^{\alpha}=-\mathrm{i}b\) & \(t^{k}\cos(bt)\) & \(\mathrm{Re}(t^{k}e^{bt})\) & \(t^{ak}t^{ak}t^{ak-1}\cos_{\alpha,\alpha}^{(k)}(t^{ak})\) & \(\mathrm{Re}(\varepsilon_{k}(t,\mathrm{i}b;\alpha,\alpha))\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Correspondence between the basis functions of IOS and FOS for various types of poles. Note that all these basis functions of FOS for multiple poles can be expressed using the corresponding derivatives of the two-parameter MLF of type \(E_{\alpha,\alpha}\). Based on defining relation (28) for [PERSON]'s function \(\varepsilon_{k}\), it is easy to prove that all basis functions for the FOS, listed in the \"Basis Functions\" column, can be expressed just using the function \(\varepsilon_{k}\) as listed in the \"Generated from\" column. Thus, Table 5 provides guidance on how to obtain the time response of a commensurate FOS from the response of the associated classical IOS, namely by substituting for each other the appropriate basis functions. The most systematic correspondence is provided by comparing the generating functions \(f(t)\) for IOS (1) and \(f_{F}(t)\) for FOS,  
\[f_{F}(t)=\varepsilon_{k}(t,a+\mathrm{i}b;\alpha,\alpha) \tag{52}\]  
(see the comparison of the \"Generated from\" columns in Table 5): for real poles of a general multiplicity, the basis functions are generated with the function \(t^{k}\)\(e^{\mu t}\) or \(\varepsilon_{k}(t,a;\alpha,\alpha)\), and for complex poles of a general multiplicity, by the real and imaginary parts of the function \(t^{k}\)\(e^{(a+\mathrm{i}b)t}\) or \(\varepsilon_{k}(t,a+\mathrm{i}b;\alpha,\alpha)\).