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This paper provides the solution to the complex-order differential equation, "0d"t^qx(t)=kx(t)+bu(t), where both q and k are complex. The time-response solution is shown to be a series that is complex-valued. Combining this system with its complex conjugate-order system yields the following generalized differential equation, "0d"t^2^R^e^(^q^)x(t)-k@?"0d"t^qx(t)-k"0d"t^q^@?x(t)+kk@?x(t)=p"0d"t^qu(t)+p@?"0d"t^q^@?u(t)-(k+k@?)u(t). The transfer function of this system is p(s^q-k)^-^1+p@?(s^q^@?-k@?)^-^1, having a time-response 2@?"n"="0^~t^(^n^+^1^)^u^-^1Repk^n@C((n+1)q)cos((n+1)vlnt)-Impk^n@C((n+1)q)sin((n+1)vlnt). The transfer function has an infinite number of complex-conjugate pole pairs. Bounds on the parameters u=Re(q),v=Im(q), and k are determined for system stability. |
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