Closed-form expressions of transfer function responses

General closed-form expressions of continuous as well as discrete time transfer-function responses have been derived. The closed-form expressions for real and/or complex - but excluding repeated eigenvalues, include the numerator coefficients of the transfer function, a Vandermonde-like matrix and a vector containing the transfer-function eigen-responses, the latter two only depending on the eigenvalues. The closed-form expressions are extended to include repeated eigenvalues as well. Such closed-form expressions lend themselves easily to analysis, direct computation of time responses, as well as to optimization and direct computation of PID controllers.