), which reduces the Laplace transforms with complex arguments to Fourier transforms with real argument ω. The applications where this is common are ones where there is interest only in the steady-state response of an LTI system, not the fleeting turn-on and turn-off behaviors or stability issues. That is usually the case for signal processing and communication theory. Thus, for continuous-time input signal and output , the transfer function is the linear mapping of the Laplace transform of the input, , to the Laplace transform of the output : or In discrete-time systems, the relation between an input signal and output is dealt with using the z-transform, and then the transfer function is similarly written as and this is often referred to as the pulse-transfer function. Direct derivation from differential equations Consider a linear differential equation with constant coefficients where u and r are suitably smooth functions of t, and L is the operator defined on the relevant function space, that transforms u into r. That kind of equation can be used to constrain the output function u in terms of the forcing function r. The transfer function can be used to define an operator that serves as a right inverse of L, meaning that . Solutions of the homogeneous, constant-coefficient differential equation can be found by trying . That substitution yields the characteristic polynomial The inhomogeneous case can be easily solved if the input function r is also of the form . In that case, by substituting one finds that if we define Taking that as the definition of the transfer function requires careful disambiguation between complex vs. real values, which is traditionally influenced by the interpretation of abs(H(s)) as the gain and −atan(H(s)) as the phase lag. Other definitions of the transfer function are used: for example Gain, transient behavior and stability A general sinusoidal input to a system of frequency may be written . The response of a system to a sinusoidal input beginning at time will consist of the sum of the steady-state response and a transient response. The steady-state response is the output of the system in the limit of infinite time, and the transient response is the difference between the response and the steady state response (it corresponds to the homogeneous solution of the above differential equation). The transfer function for an LTI system may be written as the product: where sPi are the