Transfer function
A transfer function is a mathematical representation of the relation between the input and output of a linear time-invariant system. It is mainly used in signal processing.
Background
Take a complex harmonic signal with amplitude A, angular frequency w and phase p
x(t) = A0 * exp(j * (w0 * t + p0))
and use it as an input to a system with transfer function H(j w). The output will match the following equation:
y(t) = A1 * exp(j * (w0 * t + p1)).
Note that the fundamental frequency has not changed, only the amplitude and the phase of the response changed as it went through the system. The transfer function describes for frequency w0,
A1/A0 = | H(j * w0) |
and
p1 - p0 = angle( H(j * w0) ).