Given a continuous time transfer function g(s), this paper provides closed-form solutions for the coefficients of discrete time transfer functions g(z). Similar solutions are also provided for g(s), given g(z). Three integration methods are used for each direction (termed forward and inverse, respectively) of transformation : (a) the ’forward rectangular rule’; (b) the ’backward rectangular rule’; and (c) Tustin’s method (the ’trapezoidal rule’). Thus, the normally cumbersome method of substitution for s by a rational approximation for e(sDELTA), where DELTA is the sampling interval, followed by algebraic sorting of the z- or s-domain polynomials, is completely circumvented by the methods described. Examples are given to show the simplicity and efficacy of the methods.