There are a variety of engineering applications in which the dynamics of the equipment being used or designed are expressible by a Laplace transform which contains both polynomial functions of s and exponential functions. In networks and in feedback control systems, transcendental terms in the Laplace transform arise due to diffusion in a transistor, dead time or time of flow of materials in heat exchangers and flow reactors, catalyst flow time in a catalytic cracker, time of transmission of an electromagnetic pulse in radar and telemetering applications, time of transmission of an ultrasonic pulse in sonar, materials transport time in thickness controls for rolling mills, the sampling time in sampled data systems and in digital computer controls, and the carrier period in magnetic-amplifier and thyratron-type controls. This paper will present some of the basic features of the mathematical treatment of mixed transcendental and polynomial transforms. Root locus methods, complex-plane techniques for factoring polynomials, and evaluation of the residues for an inverse transform can all be performed through the use of two complex planes, one for the polynomial representation, and the other for the transcendental terms.