Dispersion relations for third-degree nonlinear systems

A time-independent casual system is considered, in which the effect depends cubically on the cause. The system's response function depends on three time variables and its transform on three frequency variables. It is shown that there is an analog of the Kramers-Kronig dispersion relations, so that the real part of the transform can be obtained from the imaginary part by integrating over the observed frequency in a certain way, and vice versa. It is shown how the transform function is determined by the effects produced by causes built up from pure frequencies. Also it is shown how the real and imaginary parts of the transform can be separately found by time averaging the product of pure-frequency responses with the cubes of the causes.