A solution to the fundamental linear complex-order differential equation

This paper provides the solution to the complex-order differential equation, 0dtqx(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, 0dt2Re(q)x(t)-k¯0dtqx(t)-k0dtq¯x(t)+kk¯x(t)=p0dtqu(t)+p¯0dtq¯u(t)-(k+k¯)u(t). The transfer function of this system is p(sq-k)-1+p¯(sq¯-k¯)-1, having a time-response 2∑n=0∞t(n+1)u-1RepknΓ((n+1)q)cos((n+1)vlnt)-ImpknΓ((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)u=Re(q),v=Im(q), and k are determined for system stability.