General analytic expressions of transfer function responses are derived in this paper, The analytic forms include the numerator coefficients of the transfer function, a Vandermonde-like matrix, and a vector containing the transfer function eigenresponses, the latter two only depending on the eigenvalues. Severer characteristics of basic transient responses are derived, e.g., their exact number of zeros, an upper hound on the number of extrema of the step response is obtained an algorithm is suggested for the selection of the numerator coefficients, assuming fixed eigenvalues, using the derived characteristics of the basic transient responses, effectively leading to a suboptimal choice, Further, the analytic formulas are used to calculate the numerator coefficients optimally, by minimizing the step transient, the lamp transient, the parabolic transient, etc, Simulations demonstrating excellent results are presented, clearly indicating the properties of the basic transient responses and the transfer function response properties based on optimal and suboptimal choices of the numerator coefficients.