A general series expansion is derived for the amplitude of any modulation component in the output of a single-valued non-linearity subjected to an input which is the sum of any number of cosine waves of different frequencies. An important property of this expansion is that it expresses the response of the non-linearity to a multiple cosine wave input in terms of its response to a single cosine wave input; thus an inter-modulation component or a Multiple Input Describing Function (MIDF) can be obtained by using the coefficients of the Fourier series expansion of the output of the non-linearity when the input is a single cosine wave of appropriate amplitude. Since these coefficients can also be obtained by measurements, the method may be applied directly to empirical non-linearities without resorting to piecewise linear or polynomial approx imations.