Generalized exponential functions are defined and developed for the analysis and aynthesis of dynamical systems. Approximation of conventional response transforms in the 8-domain is implemented by utilizing non-integer-order complex operator 8v. The analysis technique is applied to a non-inductive long cable and to network systems. In addition, for the synthesis, Ĺaguorre-Ĺeo functions in the 8-domain are extended into the non-integer-order 8v-domain in terms of generalized exponential functions. The application is demonstrated by an example